Solving First Order Differential Equations (Edexcel A Level Further Maths: Core Pure): Revision Note

First order differential equations

What is a differential equation?

• A differential equation is simply an equation that contains derivatives

  • For example $\frac{\mathrm{d}y}{\mathrm{d}x}=12x{y}^2$ is a differential equation

  • And so is $\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}-5\frac{\mathrm{d}x}{\mathrm{d}t}+7x=5\sin t$

What is a first order differential equation?

• A first order differential equation is a differential equation that contains first derivatives but no second (or higher) derivatives

  • For example $\frac{\mathrm{d}y}{\mathrm{d}x}=12x{y}^2$ is a first order differential equation

  • But $\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}-5\frac{\mathrm{d}x}{\mathrm{d}t}+7x=5\sin t$ is not a first order differential equation, because it contains the second derivative $\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}$

• The general solution to a first order differential equation will have one unknown constant

• To find the particular solution you will need to know an initial condition or a boundary condition

Wait – haven’t I seen first order differential equations before?

• Yes you have!

  • For example $\frac{\mathrm{d}y}{\mathrm{d}x}=3x^2$ is also a first order differential equation, because it contains a first derivative and no second (or higher) derivatives

  • But for that equation you can just integrate to find the solution y = x³ + c (where c is a constant of integration)

• In A Level Maths you will have solved some first order differential equations using the method of separation of variables

Integrating factors

What is an integrating factor?

• An integrating factor can be used to solve a differential equation that can be written in the standard form $\frac{dy}{dx}+p\left(x\right)y=q\left(x\right)$

  • Be careful – the ‘functions of x’ p(x) and q(x) may just be constants!

    • For example in $\frac{dy}{dx}+6y={\mathrm{e}}^{-2x}$, p(x) = 6 and q(x) = e^-2x

    • While in $\frac{dy}{dx}+\frac{y}{2x}=12$, $p\left(x\right)=\frac{1}{2x}$  and q(x) = 12

• For an equation in standard form, the integrating factor is ${\mathrm{e}}^{\int p\left(x\right)\mathrm{d}x}$

How do I use an integrating factor to solve a differential equation?

• STEP 1: If necessary, rearrange the differential equation into standard form

• STEP 2: Find the integrating factor

  • Note that you don’t need to include a constant of integration here when you integrate  ∫p(x) dx

• STEP 3: Multiply both sides of the differential equation by the integrating factor

• This will turn the equation into an exact differential equation of the form $\frac{\mathrm{d}}{\mathrm{d}x}\left(y{\mathrm{e}}^{\int p\left(x\right)\mathrm{d}x}\right)=q\left(x\right){\mathrm{e}}^{\int p\left(x\right)\mathrm{d}x}$

• STEP 4: Integrate both sides of the equation with respect to x

  • The left side will automatically integrate to $y{\mathrm{e}}^{\int p\left(x\right)dx}$

  • For the right side, integrate $\int q\left(x\right){\mathrm{e}}^{\int p\left(x\right)dx}dx$ using your usual techniques for integration

  • Don’t forget to include a constant of integration

    • Although there are two integrals, you only need to include one constant of integration

• STEP 5: Rearrange your solution to get it in the form y = f(x)

What else should I know about using an integrating factor to solve differential equations?

• After finding the general solution using the steps above you may be asked to do other things with the solution

  • For example you may be asked to find the solution corresponding to certain initial or boundary conditions