Series solutions of differential equationsEdexcel International A Level Further Maths: Flashcards

What these 12 flashcards ask

What is the Taylor series method for a differential equation?
How many initial conditions does a second-order equation need?
How do you find y''(0) for a second-order equation?
How do you find y'''(0)?
Differentiate y''=xy.
Differentiate y^2 with respect to x.
Differentiate xy' with respect to x.
Series solution of y''=xy, y(0)=1, y'(0)=1 up to x^4?
Series solution of y'=x^2+y, y(0)=1 up to x^4?
Series for y''+xy'+y=0, y(0)=1, y'(0)=0 up to x^4?
How can you check a series solution?
Which derivatives are needed for a series up to x^4?

Exam questions on Series solutions of differential equations

The function $y(x)$ satisfies the differential equation $\frac{d^2y}{dx^2}=x+2y$ , with $y=1$ and $\frac{dy}{dx}=2$ at $x=0$ . A series solution in ascending powers of $x$ is to be found.
Find the series solution up to and including the term in $x^4$ , and use it to estimate $y(0.2)$ to 4 decimal places.2 marks
The function $y(x)$ satisfies the differential equation $\frac{d^2y}{dx^2}=xy$ , with $y=1$ and $\frac{dy}{dx}=1$ at $x=0$ .
Find the series solution up to and including the term in $x^4$ , and use it to estimate $y(0.4)$ to 3 decimal places.2 marks
The function $y(x)$ satisfies the first-order differential equation $\frac{dy}{dx}=x^2+y$ , with $y=1$ at $x=0$ .
Find the values of $\frac{dy}{dx}$ , $\frac{d^2y}{dx^2}$ and $\frac{d^3y}{dx^3}$ at $x=0$ .3 marks

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).