Advanced Higher Maths
Differential Equations

This is a simple example of a first-order, separable differential equation. We can move all the \(\raise 0.3pt{y}\) terms to one side and all the \(\raise 0.2pt{x}\) terms to the other. Then we simply integrate both sides with respect to their respective variables.

$$ \begin{gather} 3y\,\small\frac{dy}{dx}\normalsize=\small\frac{2x}{y}\normalsize \\[8pt] 3y^2\,dy=2x\,dx \\[8pt] \int 3y^2\,dy=\int 2x\,dx \\[8pt] y^3=x^2+c \\[8pt] y=\sqrt[\leftroot{-1}\uproot{6}\scriptstyle 3]{x^2+c\:} \\[8pt] \end{gather} $$

Detailed breakdown by ChatGPT 

Although this looks quite similar to the previous example and is also a first-order, separable differential equation, it is significantly more difficult.

However, the need to use partial fractions to enable integration is in line with the specification .

First we separate the variables:

$$ \begin{gather} x\,\small\frac{dy}{dx}\normalsize=y-y^2 \\[6pt] x\,dy=(y-y^2)\,dx \\[6pt] \small\frac{dy}{y-y^2}\normalsize=\small\frac{dx}{x} \\[6pt] \end{gather} $$

To enable integration, we need to express the left-hand integrand in partial fractions:

Substituting \(y\!=\!0\) gives us \(A\!=\!1\) and substituting \(y\!=\!1\) gives us \(B\!=\!1\small.\)

Now we can return to our solution. We integrate and use the laws of logarithms to simplify:

$$ \begin{gather} \int\small\frac{dy}{y-y^2}\normalsize=\int\small\frac{dx}{x}\normalsize \\[8pt] \int\left(\small\frac{1}{y}\normalsize+\small\frac{1}{1-y}\normalsize\right)dy=\int\small\frac{1}{x}\normalsize\,dx \\[10pt] \text{ln}\,\vert y\vert-\text{ln}\,\vert 1-y\vert=\text{ln}\,\vert x\vert+c \\[9pt] \text{ln}\,y-\text{ln}\,(1-y)=\text{ln}\,x+\text{ln}\,k\textsf{, say} \\[9pt] \text{ln}\left(\small\frac{y}{1-y}\normalsize\right)=\text{ln}\,kx \\[9pt] \small\frac{y}{1-y}\normalsize=kx \\[8pt] \end{gather} $$

Note that writing \(c\) as \(\text{ln}\,k\) (or some other letter) is a standard 'trick' to enable simplification.

Note also that we can remove the modulus signs because the question tells us that \(x\gt 0\) and \(0\lt y\lt 1\small.\)

The final part of obtaining the general solution is to make \(y\) the subject:

$$ \begin{gather} y=kx(1-y)\\[7pt] y=kx-kxy\\[7pt] y+kxy=kx\\[7pt] y(1+kx)=kx\\[7pt] y=\small\frac{kx}{1+kx}\\[7pt] \end{gather} $$

Detailed breakdown by ChatGPT 

This is also a first-order, separable ODE, so we start by separating \(x\) and \(y\):

$$ \begin{gather} \small\frac{dy}{dx}\normalsize = \small\frac{\text{sec}\,y}{y}\normalsize \\[8pt] y\ dy = \text{sec}\,y\ dx \\[8pt] y\,\text{cos}\,y\ dy = 1\ dx \\[8pt] \int y\,\text{cos}\,y\ dy = \int 1\ dx \\[8pt] \end{gather} $$

The left hand side is a product of two functions of \(y\small,\) so it needs integration by parts.

So we continue like this:

$$ \begin{gather} y\,\text{sin}\,y-\int 1\,.\,\text{sin}\,y\ dy = x+c \\[6pt] y\,\text{sin}\,y+\text{cos}\,y = x+c \\[6pt] \end{gather} $$

The wording of the question allows us to leave this general solution in implicit form (which is fortunate, as it isn't separable using any algebraic method!)

Now we substitute the initial conditions \(\raise 0.3pt{x\!=\!\large\frac{\pi}{4}\small,\,}\) \(\raise 0.3pt{y\!=\!\large\frac{\pi}{2}}\) to work towards the particular solution:

$$ \begin{gather} \small\frac{\pi}{2}\normalsize\,\text{sin}\,\small\frac{\pi}{2}\normalsize+\text{cos}\,\small\frac{\pi}{2}\normalsize = \small\frac{\pi}{4}\normalsize+c \\[6pt] \small\frac{\pi}{2}\normalsize\left(1\right)+0 = \small\frac{\pi}{4}\normalsize+c \\[6pt] c = \small\frac{\pi}{2}\normalsize-\small\frac{\pi}{4}\normalsize \\[6pt] c = \small\frac{\pi}{4}\normalsize \\[6pt] \end{gather} $$

So the particular solution is:

Detailed breakdown by ChatGPT 

This isn't separable. It is a relatively simple example of a first-order linear differential equation. It is in the form:

In this example:

We will show you two methods of solution. Both of them rely on the integrating factor (IF), defined as follows:

$$ \begin{eqnarray} I(x) &=& e\,^{\large\int\normalsize P(x)\,dx} \\[6pt] &=& e\,^{\large\int\normalsize 2\,dx} \\[6pt] &=& e^{2x} \\[6pt] \end{eqnarray} $$

Note that we do not need to worry about any constant of integration in the IF.

Method 1: Reverse product rule

Multiply both sides by the IF. The left hand side will take on the form of a product that has been differentiated.

$$ \begin{gather} \small\frac{dy}{dx}\normalsize+2y=5e^{3x} \\[8pt] \underbrace{e^{2x}}_{v}\,\small\underbrace{\frac{dy}{dx}}_{u'}\normalsize+\underbrace{2e^{2x}}_{v'}\underbrace{y}_{u}=5e^{5x} \\[10pt] \small\frac{d}{dx}\normalsize\left(y\small\,\normalsize e^{2x}\right)=5e^{5x} \\[8pt] y\small\,\normalsize e^{2x}=\int\!5e^{5x}\,dx \\[8pt] y\small\,\normalsize e^{2x}=e^{5x}+c \\[8pt] y=e^{3x}+\small\frac{c}{e^{2x}} \\[8pt] \end{gather} $$

Method 2: A handy shortcut

The first method is totally logical but a bit laborious. There is a shortcut that can be proven to be an equivalent method, but when first seen it often induces a "Huh?! Where did that come from?" reaction. Anyway, here it is:

If you are happy to take this on trust and learn the shortcut, it can save a little time. Here goes:

$$ \begin{gather} I(x)\small\,\normalsize y=\int I(x)\small\,\normalsize Q(x)\,dx \\[6pt] e^{2x}\small\,\normalsize y=\int e^{2x}\,.\,5e^{3x}\,dx \\[6pt] e^{2x}\small\,\normalsize y=\int\!5e^{5x}\,dx \\[6pt] e^{2x}\small\,\normalsize y=e^{5x}+c \\[6pt] y=e^{3x}+\small\frac{c}{e^{2x}} \\[6pt] \end{gather} $$

Detailed breakdown by ChatGPT 

This is another first-order linear differential equation. However, to put it into the standard form we first need to divide through by \(\raise 0.2pt{x}\small.\)

Now we can compare this with the standard form of a first-order linear ODE:

So we can see that:

Like the previous example, we will show you both methods of obtaining the general solution. You may decide which suits you better. But first, we need the integrating factor:

$$ \begin{eqnarray} I(x) &=& e\,^{\large\int\normalsize P(x)\,dx} \\[6pt] &=& e\,^{\large\int\frac{2}{x}\normalsize\,dx} \\[6pt] &=& e\,^{\large 2\,\text{ln}\,x} \\[6pt] &=& e\,^{\large\text{ln}\,{x^2}} \\[6pt] &=& x^2 \\[6pt] \end{eqnarray} $$

Method 1: Reverse product rule

Multiply through by the IF and use the product rule in reverse.

In this example, the right hand side needs integration by parts.

$$ \begin{gather} \small\frac{dy}{dx}\normalsize+\small\frac{2y}{x}\normalsize=\small\frac{\text{cos}\,x}{x}\normalsize \\[7pt] x^{2}\small\,\frac{dy}{dx}\normalsize+2xy=x\,\text{cos}\,x \\[6pt] \small\frac{d}{dx}\normalsize\left(x^{2}y\right)=x\,\text{cos}\,x \\[6pt] x^{2}y=\int x\,\text{co}s\,x\ dx \\[4pt] x^{2}y=x\,\text{sin}\,x-\int 1\,.\,\text{sin}\,x\ dx \\[6pt] x^{2}y=x\,\text{sin}\,x+\text{cos}\,x+c \\[9pt] y=\small\frac{\text{sin}\,x}{x}\normalsize+\small\frac{\text{cos}\,x}{x^2}\normalsize+\small\frac{c}{x^2}\normalsize \\[6pt] \end{gather} $$

Method 2: The shortcut

This isn't much shorter. Its only slight advantage is that you don't have to think about the product rule. We prefer the first method, but you make up your own mind.

$$ \begin{gather} I(x)\small\,\normalsize y=\int I(x)\small\,\normalsize Q(x)\,dx \\[6pt] x^{2}y=\int x^2\,.\,\small\frac{\text{cos}\,x}{x}\normalsize\,dx \\[6pt] x^{2}y=\int x\,\text{cos}\,x\,dx \\[4pt] x^{2}y=x\,\text{sin}\,x-\int\ 1\,.\,\text{sin}\,x\ dx \\[6pt] x^{2}y=x\,\text{sin}\,x+\text{cos}\,x+c \\[9pt] y=\small\frac{\text{sin}\,x}{x}\normalsize+\small\frac{\text{cos}\,x}{x^2}\normalsize+\small\frac{c}{x^2}\normalsize \\[6pt] \end{gather} $$

Detailed breakdown by ChatGPT 

This is our first example of a second-order linear differential equation.

Because the right hand side is zero we call this homogeneous.

The general form of a homogeneous second-order linear differential equation is:

Our first step is to form the auxiliary equation from the left hand side:

$$ \begin{gather} am^2+bm+c=0 \\[6pt] m^2-3m-10=0 \\[6pt] (m+2)(m-5)=0 \\[6pt] m=-2\:\ \small\textsf{or}\normalsize\:\ m=5 \\[6pt] \end{gather} $$

Note that we have two distinct, real roots of the auxiliary equation. That's important. When we have real and distinct roots \(\raise 0.2pt{p}\) and \(\raise 0.2pt{q}\small,\) the general solution of the differential equation is always of the form:

where \(A\) and \(B\) are constants.

So in this example, the general solution is:

Now we turn our attention to using the initial conditions \(\raise 0.2pt{x\!=\!0}\small,\) \(\raise 0.3pt{y\!=\!2\small,}\) \(\large\frac{dy}{dx}\normalsize\!=\!-11\) to find the particular solution.

First we need to differentiate the general solution to obtain an expression for \(\large\frac{dy}{dx}\small.\)

Now we substitute the initial conditions into our two equations and solve simultaneously:

So \(A\!=\!3\small\) giving \(B\!=\!-1\) and the particular solution:

Detailed breakdown by ChatGPT 

Like the previous example, this is a second-order linear homogeneous differential equation. The general form is:

So we solve the auxiliary equation:

$$ \begin{gather} am^2+bm+c=0 \\[6pt] 9m^2-12m+4=0 \\[6pt] (3m-2)(3m-2)=0 \\[5pt] m=\small\frac{2}{3}\normalsize \\[6pt] \end{gather} $$

Note that we have real and equal roots of the auxiliary equation. That's important. When we have a repeated real root, \(\raise 0.2pt{p}\) say, the general solution of the differential equation is always of the form:

where \(A\) and \(B\) are constants.

So in this example, the general solution is:

Note that this may also be expressed in factorised form:

Detailed breakdown by ChatGPT 

Like example 6 and example 7, this is a second-order linear homogeneous differential equation.

So we solve the auxiliary equation, which in this case does not factorise:

$$ \begin{gather} am^2+bm+c=0 \\[6pt] m^2+2m+5=0 \\[6pt] m=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\[6pt] m=\frac{-2\pm\sqrt{4-20}}{2} \\[6pt] m=\frac{-2\pm\sqrt{-16}}{2} \\[6pt] m=\frac{-2\pm4i}{2} \\[6pt] m=-1\pm 2i \\[6pt] \end{gather} $$

Note that we have complex conjugate roots of the auxiliary equation. That's important. When we have complex conjugate roots \(\raise 0.2pt{p\pm qi\small,}\) the general solution of the differential equation is always of the form:

where \(A\) and \(B\) are constants.

In this example, \(p\!=\!-1\) and \(q\!=\!2\) so the general solution is:

Detailed breakdown by ChatGPT 

Like the previous three examples, this is a second-order linear differential equation, but this time the right hand side is not zero, so it is non-homogeneous.

We start by solving the related homogeneous equation. Its general solution is known as the complementary function, and it forms part of the general solution of the original non-homogeneous equation.

The related homogeneous equation is:

The auxiliary equation (AE) is:

$$ \begin{gather} m^2-5m+4=0 \\[6pt] (m-1)(m-4)=0 \\[6pt] m=1\:\ \small\textsf{or}\normalsize\:\ m=4 \\[6pt] \end{gather} $$

The AE has distinct, real roots so the complementary function (CF) is:

where \(A\) and \(B\) are constants.

Now we focus on the right hand side of the original ODE: \(\raise 0.2pt{4x-1\small.}\) This is a linear expression so a particular solution of the original equation will be in the form \(y=Cx+D\) where \(\raise 0.1pt{C}\) and \(D\) are constants which we can determine. We start by differentiating twice:

$$ \begin{gather} y=Cx+D \\[6pt] \small\frac{dy}{dx}\normalsize =C \\[6pt] \small\frac{d^{2}y}{dx^2}\normalsize =0 \\[6pt] \end{gather} $$

Now we substitute these back into the original differential equation:

$$ \begin{gather} \small\frac{d^{2}y}{dx^2}\normalsize-5\small\frac{dy}{dx}\normalsize+4y=4x-1 \\[8pt] 0-5C+4(Cx+D)=4x-1 \\[8pt] -5C+4Cx+4D=4x-1 \\[8pt] 4Cx+(4D-5C)=4x-1 \\[8pt] \end{gather} $$

Equating the coefficients of \(\raise 0.2pt{x}\):

$$ \begin{gather} 4C=4 \\[6pt] C=1 \\[6pt] \end{gather} $$

Equating the constant coefficients:

$$ \begin{gather} 4D-5C=-1 \\[6pt] 4D-5=-1 \\[6pt] 4D=4 \\[6pt] D=1 \\[6pt] \end{gather} $$

What we call the particular integral (PI) is therefore \(\raise 0.2pt{Cx+D=x+1\small.}\)

Finally, the general solution of the original equation is just the complementary function plus the particular integral:

$$ y=Ae^{x}+Be^{4x}+x+1 $$

Detailed breakdown by ChatGPT 

Like the previous example, this is a non-homogeneous second-order linear differential equation, although this example is going to have an added complication, as you will see below. Deep breath...

We start by solving the related homogeneous equation:

The auxiliary equation (AE) is:

$$ \begin{gather} m^2-4m+4=0 \\[6pt] (m-2)(m-2)=0 \\[6pt] m=2\\[6pt] \end{gather} $$

Because the AE has a repeated real root, the complementary function (CF) is:

where \(A\) and \(B\) are constants.

Now we consider the right hand side of the original equation: \(\raise 0.2pt{6e^{2x}\small.}\) Because it is an exponential expression we would usually try \(\raise 0.2pt{y=Ce^{2x}}\) (where \(\raise 0.2pt{C}\) is a constant) as the particular integral (PI).

But here is where the complication arises: there is already a term of that form in the CF: the \(\raise 0.2pt{Ae^{2x}\small.}\) When that happens, we multiply by \(\raise 0.2pt{x}\small,\) giving \(y=Cxe^{2x}\small.\) But that's no good either, because there is also a term of that form in the CF: the \(\raise 0.2pt{Bxe^{2x}\small.}\) When this happens, we need to multiply by \(\raise 0.2pt{x}\) again, so now we have \(\raise 0.2pt{y=Cx^{2}e^{2x}\small.}\)

Now we use the product rule to find the first and second derivatives. Check these for yourself; they're fiddly!

$$ \begin{eqnarray} y&=&Cx^{2}e^{2x} \\[6pt] \small\frac{dy}{dx}\normalsize &=&2Cx^{2}e^{2x}+2Cxe^{2x} \\[6pt] \small\frac{d^{2}y}{dx^2}\normalsize &=& 4Cx^{2}e^{2x}+8Cxe^{2x}+2Ce^{2x}\\[6pt] \end{eqnarray} $$

Next, we substitute these back into the original differential equation:

$$ \begin{gather} \small\frac{d^{2}y}{dx^2}\normalsize-4\small\frac{dy}{dx}\normalsize+4y=6e^{2x}\\[10pt] \end{gather} $$

$$ \begin{eqnarray} 4Cx^{2}e^{2x}\!+\!8Cxe^{2x}\!+\!2Ce^{2x}\:\:\:\:\:\:\:\:\:\:\:\:&&\: \\[6pt] -8Cx^{2}e^{2x}-8Cxe^{2x}+4Cx^{2}e^{2x}&=&6e^{2x} \\[10pt] \end{eqnarray} $$

Thankfully we don't need to go any further with this beastly equation! Its sole purpose was to let us find \(\raise 0.2pt{C\small.}\)

So, equating the coefficients of \(\raise 0.2pt{e^{2x}\small,}\) we obtain \(\raise 0.2pt{2C=6\small,}\) so \(\raise 0.2pt{C=3}\) and the particular integral (PI) is \(\raise 0.2pt{3x^{2}e^{2x}\small.}\)

Finally, the general solution of the original equation is just the complementary function plus the particular integral:

$$ y=Ae^{2x}+Bxe^{2x}+3x^{2}e^{2x} $$

Note: A modified PI has only been required in one recent exam paper: 2021 P1 Q8 . See example 13 below.

Detailed breakdown by ChatGPT 

Like the previous two examples, this is a non-homogeneous second-order linear differential equation, although in this example we need to go further and provide a particular solution, not just the general solution.

Questions of this type, usually worth 9 or 10 marks, appear quite often on Advanced Higher exam papers.

We start by solving the related homogeneous ODE:

The auxiliary equation (AE) is:

$$ \begin{gather} m^2-6m+9=0 \\[6pt] (m-3)^2=0 \\[6pt] m=3\\[6pt] \end{gather} $$

Because the AE has a repeated real root, the complementary function (CF) is:

where \(A\) and \(B\) are constants.

Now we consider the right hand side of the original equation: \(8\,\text{sin}\,x+19\,\text{cos}\,x\small.\)

Because it is a trigonometic expression we will use \(C\,\text{sin}\,x+D\,\text{cos}\,x\) (where \(\raise 0.2pt{C}\) and \(\raise 0.2pt{D}\) are constants) as the particular integral.

So, differentiating twice:

$$ \begin{eqnarray} y &=& C\,\text{sin}\,x+D\,\text{cos}\,x \\[6pt] \small\frac{dy}{dx}\normalsize &=& C\,\text{cos}\,x-D\,\text{sin}\,x \\[6pt] \small\frac{d^{2}y}{dx^2}\normalsize &=& -\!C\,\text{sin}\,x-D\,\text{cos}\,x \\[6pt] \end{eqnarray} $$

Next, we substitute these back into the original differential equation:

$$ \begin{gather} \small\frac{d^{2}y}{dx^2}\normalsize-6\small\frac{dy}{dx}\normalsize+9y=8\,\text{sin}\,x+19\,\text{cos}\,x\\[10pt] \end{gather} $$

For clarity, let's split the substitution over several lines:

$$ \begin{gather} -C\,\text{sin}\,x-D\,\text{cos}\,x \\[6pt] -6(C\,\text{cos}\,x-D\,\text{sin}\,x) \\[6pt] +9(C\,\text{sin}\,x+D\,\text{cos}\,x) \\[6pt] \:= 8\,\text{sin}\,x+19\,\text{cos}\,x \\[10pt] \end{gather} $$

Equating the coefficients of \(\text{sin}\,x\):

$$ \begin{eqnarray} -C+6D+9C &=& 8 \\[6pt] 8C+6D &=& 8 \\[6pt] 4C+3D &=& 4\:\:① \\[6pt] \end{eqnarray} $$

Equating the coefficients of \(\text{cos}\,x\):

$$ \begin{eqnarray} -D-6C+9D &=& 19 \\[6pt] -6C+8D &=& 19\:\:② \\[6pt] \end{eqnarray} $$

Now we are ready to solve equations \(①\) and \(②\) simultaneously:

$$ \begin{eqnarray} 3\times①:&&\:\:\:12C &+& 9D &=& 12 && \:\:③\\[6pt] 2\times②:&&-\!12C &+& 16D &=& 38 && \:\:④\\[6pt] \end{eqnarray} $$

Eliminating \(\raise 0.2pt{C}\):

$$ \begin{eqnarray} ③+④:\:\:25D &=& 50\\[6pt] D &=& 2\\[6pt] \end{eqnarray} $$

Substituting into \(①\) to find \(\raise 0.2pt{C}\):

$$ \begin{gather} 4C+3D=4\\[6pt] 4C+6=4\\[6pt] 4C=-2\\[6pt] C=-\small\frac12\normalsize \\[6pt] \end{gather} $$

So, adding the CF and PI, we have the following:

$$ y=Ae^{3x}+Bxe^{3x}-\small\frac12\normalsize\,\text{sin}\,x+2\,\text{cos}\,x $$

Finally, we need to use the given initial conditions to find \(\raise 0.2pt{A}\) and \(\raise 0.2pt{B}\small.\)

Substituting \(y\!=\!7\) and \(x\!=\!0\):

$$ \begin{gather} 7 =Ae^{0}+B(0)e^{0}-\small\frac12\normalsize\,\text{sin}\,0+2\,\text{cos}\,0\\[8pt] 7 = A+0-0+2\\[8pt] A = 5\\[8pt] \end{gather} $$

To find \(\raise 0.2pt{B,}\) we need to differentiate and then substitute \(\large\frac{dy}{dx}\normalsize\!=\!\large\frac12\normalsize\) and \(x\!=\!0\):

$$ \begin{gather} \small\frac{dy}{dx}\normalsize\!=\!3Ae^{3x}+Be^{3x}(3x\!+\!1)-\small\frac12\normalsize\text{cos}\,x-2\,\text{sin}\,x\\[8pt] \small\frac12\normalsize=15e^0+Be^{0}(0+1)-\small\frac12\normalsize(1)-2(0)\\[8pt] \small\frac12\normalsize=15+B-\small\frac12\normalsize\\[8pt] 1=15+B\\[8pt] B=-14\\[8pt] \end{gather} $$

Putting this all together, we have the particular solution:

$$ y=5e^{3x}-14xe^{3x}-\small\frac12\normalsize\,\text{sin}\,x+2\,\text{cos}\,x $$

Detailed breakdown by ChatGPT 

This is first-order, separable, so we separate the variables and integrate:

$$ \begin{gather} \small\frac{dV}{dt}\normalsize=k(12-V)\\[8pt] dV=k(12-V)\,dt\\[8pt] \small\frac{1}{12-V}\normalsize\,dV=k\,dt\\[8pt] \displaystyle\int\small\frac{1}{12-V}\normalsize\ dV=\displaystyle\int k\ dt\\[8pt] -\text{ln}\,\vert 12-V\vert=kt+c\\[8pt] \end{gather} $$

We are told that \(V=2\) when \(t=0\small,\) so:

$$ \begin{gather} -\text{ln}\,\vert 12-2\vert =k(0)+c\\[6pt] -\text{ln}\,10 =0+c\\[6pt] c=-\text{ln}\,10\\[6pt] \end{gather} $$

As \(0\leqslant V\lt 12\small,\) we know that \(12-V\gt 0\small,\) so we can drop the modulus signs and proceed:

$$ \begin{gather} -\text{ln}(12-V)=kt-\text{ln}\,10\\[6pt] \text{ln}(12-V)=\text{ln}\,10-kt\\[6pt] e^{\large\text{ln}(12-V)\normalsize}=e^{\large\text{ln}\,10\,-\,kt}\\[6pt] 12-V=e^{\large\text{ln}\,10\normalsize}\,.\,\,e^{-kt}\\[6pt] 12-V=10e^{-kt}\\[6pt] V=12-10e^{-kt}\\[6pt] \end{gather} $$

Detailed breakdown by ChatGPT 

This 9-mark question is about as tough as this topic can get! Note that, like example 10 above, this solution needs a modified PI to ensure that it is linearly independent of the CF.

We start by solving the related homogeneous ODE:

The auxiliary equation (AE) is:

$$ \begin{gather} m^2+m-6=0 \\[6pt] (m-2)(m+3)=0 \\[6pt] m=2\:\ \small\textsf{or}\normalsize\:\ m=-3 \\[6pt] \end{gather} $$

The AE has distinct, real roots so the complementary function (CF) is:

where \(A\) and \(B\) are constants.

Now we consider the right hand side of the original equation: \(35e^{2x}\small.\) Because it is an exponential expression we would normally use \(\raise 0.2pt{y=Ce^{2x}}\) (where \(\raise 0.2pt{C}\) is a constant) as the particular integral (PI).

However, we already have a term of that form in the CF: the \(\raise 0.2pt{Ae^{2x}\small.}\) So we modify it by multiplying by \(\raise 0.2pt{x\small,}\) using \(y=Cxe^{2x}\) as the PI.

Now we use the product rule to find the first and second derivatives of the PI.

$$ \begin{eqnarray} y &=& Cxe^{2x} \\[12pt] \small\frac{dy}{dx}\normalsize &=& Ce^{2x}+2Cxe^{2x} \\[12pt] \small\frac{d^{2}y}{dx^2}\normalsize &=& 2Ce^{2x}+2Ce^{2x}+4Cxe^{2x}\\[6pt] &=& 4Ce^{2x}+4Cxe^{2x}\\[6pt] \end{eqnarray} $$

Next, we substitute these back into the original differential equation:

$$ \begin{gather} \small\frac{d^{2}y}{dx^2}\normalsize+\small\frac{dy}{dx}\normalsize-6y=35e^{2x} \\[10pt] \end{gather} $$

For clarity, let's split the substitution over multiple lines:

$$ \begin{gather} 4Ce^{2x}+4Cxe^{2x} \\[6pt] +Ce^{2x}+2Cxe^{2x} \\[6pt] -6Cxe^{2x} \\[6pt] \:= 35e^{2x} \\[10pt] \end{gather} $$

There is no need to simplify this, as its only purpose is for us to solve for \(C\small.\)

Equating the coefficients of \(e^{2x}\):

$$ \begin{eqnarray} 4C+C &=& 35 \\[6pt] 5C &=& 35 \\[6pt] C &=& 7 \\[6pt] \end{eqnarray} $$

So, adding the CF and PI, our general solution is:

$$ y = Ae^{2x}+Be^{-3x}+7xe^{2x} $$

In order to work towards finding the particular solution, we need to differentiate:

$$ \small\frac{dy}{dx}\normalsize\!= 2Ae^{2x}-3Be^{-3x}+7e^{2x}+14xe^{2x} $$

Now we can use the given initial conditions to find \(\raise 0.2pt{A}\) and \(\raise 0.2pt{B}\).

Substituting \(y\!=\!5\) and \(x\!=\!0\):

$$ \begin{gather} 5 = Ae^{0}+Be^{0}+7(0)e^{0}\\[6pt] 5 = A+B\\[6pt] \end{gather} $$

Substituting \(\frac{dy}{dx}\!=12\) and \(x\!=\!0\):

$$ \begin{gather} \small\frac{dy}{dx}\normalsize\!= 2Ae^{2x}-3Be^{-3x}+7e^{2x}+14xe^{2x}\\[6pt] 12 = 2Ae^{0}-3Be^{0}+7e^{0}+14(0)e^{0}\\[6pt] 12 = 2A-3B+7\\[6pt] 2A-3B=5\\[6pt] \end{gather} $$

Solving these two equations simultaneously by substituting the first, in the form \(A\!=\!5\!-\!B\small,\) into the second:

$$ \begin{gather} 2A-3B=5\\[6pt] 2(5\!-\!B)-3B=5\\[6pt] 10-2B-3B=5\\[6pt] 10-5B=5\\[6pt] 5B=5\\[6pt] B=1\\[6pt] \end{gather} $$

$$ y=4e^{2x}+e^{-3x}+7xe^{2x} $$

Detailed breakdown by ChatGPT 

(a)  We use the chain rule, thus:

$$ \begin{eqnarray} y &=& x\,ln\,x-x\\[6pt] \small\frac{dy}{dx}\normalsize &=& x\,.\,\small\frac{1}{x}\normalsize +ln\,x-1\\[6pt] &=& 1+ln\,x-1\\[6pt] &=& ln\,x\\[6pt] \end{eqnarray} $$

(b)  This differential equation is in the form

So it is first-order linear, non-separable.

Like example 4 and example 5 above, we can either use the reverse product rule or the shortcut method, both of which make use of the integrating factor (IF):

$$ \begin{eqnarray} I(x) &=& e\,^{\large\int ln\,x\,dx} \\[6pt] &=& e\,^{\large x\,ln\,x-x}\:\:\:\small\textsf{from part (a)}\\[6pt] \end{eqnarray} $$

Before we continue, it is worth simplifying the IF, as follows:

$$ \begin{eqnarray} I(x) &=& e\,^{\large x\,ln\,x-x} \\[6pt] &=& e\,^{\large x\,ln\,x\normalsize}\,.\,e^{\large{-x}} \\[6pt] &=& e\,^{\large ln\,{x^x\normalsize}}\,.\,e^{\large{-x}}\\[6pt] &=& x^{\large{x}}\,e^{\large{-x}}\\[6pt] \end{eqnarray} $$

Now we are ready to solve our equation. We will use the shortcut method below, but you might prefer to use the reverse product rule in full.

$$ \begin{eqnarray} I(x)\small\,\normalsize y &=& \!\int I(x)\small\,\normalsize Q(x)\,dx \\[8pt] x^{\large{x}}\,e^{\large{-x}}\,.\,y &=& \!\int x^{\large{x}}\,e^{\large{-x}}\,x^{\large{-x}}\,dx \\[8pt] &=& \!\int e^{\large{-x}}\,dx \\[8pt] &=& -\!e^{\large{-x}}+c \\[8pt] y &=& \frac{-e^{\large{-x}}+c }{x^{\large{x}}\,e^{\large{-x}}}\\[8pt] \end{eqnarray} $$

Detailed breakdown by ChatGPT 

This is a second-order linear non-homogeneous differential equation.

We start by solving the related homogeneous equation to find the complementary function:

The related homogeneous equation is:

The auxiliary equation (AE) is:

$$ \begin{gather} m^2-4m-5=0 \\[6pt] (m-5)(m+1)=0 \\[6pt] m=5\:\ \small\textsf{or}\normalsize\:\ m=-1 \\[6pt] \end{gather} $$

The AE has distinct, real roots so the complementary function (CF) is:

where \(A\) and \(B\) are constants.

Now we focus on the right hand side of the original ODE: \(\raise 0.2pt{10x^2+11x-23\small.}\) This is a quadratic expression so a particular solution of the original equation will be in the form \(y=Cx^2+Dx+E\) where \(\raise 0.1pt{C}\small,\) \(D\) and \(E\) are constants which we can determine. We start by differentiating twice:

$$ \begin{gather} y=Cx^2+Dx+E \\[6pt] \small\frac{dy}{dx}\normalsize =2Cx+D \\[6pt] \small\frac{d^{2}y}{dx^2}\normalsize =2C \\[6pt] \end{gather} $$

Now we substitute these back into the original differential equation:

$$ \begin{gather} \small\frac{d^{2}y}{dx^2}\normalsize-4\small\frac{dy}{dx}\normalsize-5y=10x^2+11x-23 \\[8pt] 2C-4(2Cx+D)-5(Cx^2+Dx+E)=10x^2+11x-23 \\[8pt] 2C-8Cx-4D-5Cx^2-5Dx-5E=10x^2+11x-23 \\[8pt] -5Cx^2+(-8C-5D)x+(2C-4D-5E)=10x^2+11x-23 \\[8pt] \end{gather} $$

Equating the coefficients of \(\raise 0.2pt{x^2}\):

$$ \begin{gather} -5C=10 \\[6pt] C=-2 \\[6pt] \end{gather} $$

Equating the coefficients of \(\raise 0.2pt{x}\):

$$ \begin{gather} -8C-5D=11 \\[6pt] -8(-2)-5D=11 \\[6pt] 16-5D=11 \\[6pt] -5D=-5 \\[6pt] D=1 \\[6pt] \end{gather} $$

Equating the constant coefficients:

$$ \begin{gather} 2C-4D-5E=-23 \\[6pt] 2(-2)-4(1)-5E=-23 \\[6pt] -4-4-5E=-23 \\[6pt] -8-5E=-23 \\[6pt] -5E=-15 \\[6pt] E=3 \\[6pt] \end{gather} $$

The particular integral (PI) is therefore \(\raise 0.2pt{-2x^2+x+3\small.}\)

Finally, the general solution of the original equation is just the complementary function plus the particular integral:

$$ y=Ae^{5x}+Be^{-x}-2x^2+x+3 $$

Differentiating to prepare for applying the initial condition for \(\large\frac{dy}{dx}\normalsize\):

$$ \small\frac{dy}{dx}\normalsize = 5Ae^{5x}-Be^{-x}-4x+1 $$

Substitute \(x=0\) and \(y=2\) into the general solution:

$$ \begin{gather} 2=Ae^{0}+Be^{0}-2(0)^2+0+3 \\[6pt] 2=A+B+3 \\[6pt] A+B=-1 \quad ① \end{gather} $$

Substitute \(x\!=\!0\) and \(\large\frac{dy}{dx}\normalsize\!=\!14\):

$$ \begin{gather} 14=5Ae^{0}-Be^{0}-4(0)+1 \\[6pt] 14=5A-B+1 \\[6pt] 5A-B=13 \quad ② \end{gather} $$

Adding equations \(①\) and \(②\):

$$ \begin{gather} 6A=12 \\[6pt] A=2 \end{gather} $$

Substituting \(A=2\) into \(①\) gives \(2+B=-1\) so \(B\!=\!-3\small.\)

Finally, substituting \(A\!=\!2\) and \(B\!=\!-3\) gives the particular solution:

$$ y=2e^{5x}-3e^{-x}-2x^2+x+3 $$

Detailed breakdown by ChatGPT 

This is first-order, separable, so we separate the variables and integrate:

$$ \begin{gather} (m-220)\small\frac{dP}{dm}\normalsize=1.4P\\[10pt] \small\frac{1}{P}\normalsize\,dP=\small\frac{1.4}{m-220}\normalsize\,dm\\[10pt] \displaystyle\int\small\frac{1}{P}\normalsize\ dP=\displaystyle\int\small\frac{1.4}{m-220}\normalsize\ dm\\[10pt] \text{ln}\,\vert P\vert=1.4\,\text{ln}\,\vert m-220\vert+c\\[10pt] \end{gather} $$

We are told that a jump of \(807\) centimetres scores \(1079\) points, so \(P\!=\!1079\) when \(m\!=\!807\small:\)

$$ \begin{gather} \text{ln}\,\vert 1079\vert =1.4\,\text{ln}\,\vert 807-220\vert+c\\[8pt] \text{ln}\,1079 =1.4\,\text{ln}\,587+c\\[8pt] c = \text{ln}\,1079 - 1.4\,\text{ln}\,587\\[8pt] c = -1.94124...\\[8pt] \end{gather} $$

As \(m>220\) and points scored \(P>0\small,\) we can drop the modulus signs and proceed:

$$ \begin{gather} \text{ln}\,P=1.4\,\text{ln}(m-220)-1.941\\[8pt] e^{\large\text{ln}\,P\normalsize}=e^{\large 1.4\,\text{ln}(m-220)\,-\,1.941\normalsize}\\[8pt] P=e^{-1.941}\,e^{\large 1.4\,\text{ln}(m-220)\normalsize}\\[8pt] P=e^{-1.941}\,e^{\large\text{ln}\left((m-220)^{1.4}\right)\normalsize}\\[8pt] P=0.144(m-220)^{1.4}\\[8pt] \end{gather} $$

Note: It is a general principle in mathematics that exactness is preferable to approximation, but in this solution the rounding is reasonable, because the exact value would be needlessly complicated.

Detailed breakdown by ChatGPT 

This is a first-order linear differential equation, already in standard form:

Comparing terms, we have:

First, we find the integrating factor:

$$ \begin{eqnarray} I(x) &=& e\,^{\large\int\normalsize P(x)\,dx} \\[6pt] &=& e\,^{\large\int -\frac{2}{x}\normalsize\,dx} \\[6pt] &=& e\,^{\large -2\,\text{ln}\,x} \\[6pt] &=& e\,^{\large\text{ln}\,{x^{-2}}} \\[6pt] &=& x^{-2} \\[6pt] &=& \small\frac{1}{x^2}\normalsize \\[6pt] \end{eqnarray} $$

Method 1: Reverse product rule

Multiply through by the IF and use the product rule in reverse.

$$ \begin{gather} \small\frac{1}{x^2}\normalsize\small\frac{dy}{dx}\normalsize-\small\frac{2}{x^3}\normalsize y=\text{sec}^2\,3x \\[7pt] \small\frac{d}{dx}\normalsize\left(\small\frac{1}{x^2}\normalsize y\right)=\text{sec}^2\,3x \\[6pt] \small\frac{1}{x^2}\normalsize y=\int\text{sec}^2\,3x\ dx \\[6pt] \small\frac{1}{x^2}\normalsize y=\small\frac{1}{3}\normalsize\text{tan}\,3x+c \\[9pt] y=x^2\left(\small\frac{1}{3}\normalsize\text{tan}\,3x+c\right) \\[6pt] \end{gather} $$

Method 2: The shortcut

$$ \begin{gather} I(x)\small\,\normalsize y=\int I(x)\small\,\normalsize Q(x)\,dx \\[6pt] \small\frac{1}{x^2}\normalsize y=\int\small\frac{1}{x^2}\normalsize\,.\,x^2\,\text{sec}^2\,3x\,dx \\[6pt] \small\frac{1}{x^2}\normalsize y=\int\text{sec}^2\,3x\,dx \\[6pt] \small\frac{1}{x^2}\normalsize y=\small\frac{1}{3}\normalsize\text{tan}\,3x+c \\[9pt] y=x^2\left(\small\frac{1}{3}\normalsize\text{tan}\,3x+c\right) \\[6pt] \end{gather} $$

Detailed breakdown by ChatGPT 

This is a homogeneous second-order linear differential equation.

First, we form and solve the auxiliary equation:

$$ \begin{gather} 2m^2-3m+1=0 \\[6pt] (2m-1)(m-1)=0 \\[6pt] m=\small\frac{1}{2}\normalsize\:\ \small\textsf{or}\normalsize\:\ m=1 \\[6pt] \end{gather} $$

Since we have two real and distinct roots, the general solution is of the form:

So the general solution is:

To use the initial conditions \(x\!=\!0\small,\) \(y\!=\!2\small,\) \(\large\frac{dy}{dx}\normalsize\!=\!-1\small,\) we first differentiate the general solution:

Substituting the initial conditions into our two equations gives:

This gives \(A\!=\!6\small.\) Substituting into \(①\) gives \(6+B=2\) so \(B\!=\!-4\small.\)

Finally, the general solution is just written with these constant values to give the particular solution:

Detailed breakdown by ChatGPT