What does a negative discriminant tell you?

The roots are complex, so the solution oscillates (with e^{px} growth or decay).

General solution of y''+ay'+by=f(x)?

Complementary function plus a particular integral.

Trial PI for f(x)=p+qx+cx^2?

y=ax^2+bx+c (a general quadratic)

What if the trial PI already appears in the CF?

Multiply the trial function by x.

When do you apply the initial conditions?

To the full general solution (CF + PI).

Second-order linear differential equationsEdexcel A-Level Further Maths: Flashcards

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Auxiliary equation for $y''+ay'+by=0$?

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Auxiliary equation for $y''+ay'+by=0$ ?: $m^2+am+b=0$
General solution for two distinct real roots $\alpha,\beta$ ?: $y=Ae^{\alpha x}+Be^{\beta x}$
General solution for a repeated root $\alpha$ ?: $y=(A+Bx)e^{\alpha x}$
General solution for complex roots $p\pm q\mathrm{i}$ ?: $y=e^{px}(A\cos qx+B\sin qx)$
What does a negative discriminant tell you?: The roots are complex, so the solution oscillates (with $e^{px}$ growth or decay).
How many arbitrary constants in the general solution of a second-order equation?: Two.
General solution of $y''+ay'+by=f(x)$ ?: Complementary function plus a particular integral.
Trial PI for $f(x)=ke^{px}$ ?: $y=\lambda e^{px}$
Trial PI for $f(x)=p+qx+cx^2$ ?: $y=ax^2+bx+c$ (a general quadratic)
Trial PI for $f(x)=m\cos\omega x+n\sin\omega x$ ?: $y=a\cos\omega x+b\sin\omega x$
What if the trial PI already appears in the CF?: Multiply the trial function by $x$ .
Roots and solution of $y''+16y=0$ ?: $m=\pm4\mathrm{i}$ , so $y=A\cos4x+B\sin4x$ .
When do you apply the initial conditions?: To the full general solution (CF + PI).

Exam questions on Second-order linear differential equations

Consider the differential equation $\frac{d^2y}{dx^2}+2\frac{dy}{dx}-8y=0$ .
Given that $y=0$ and $\frac{dy}{dx}=6$ when $x=0$ , find the particular solution.2 marks
Consider the differential equation $\frac{d^2y}{dx^2}+4\frac{dy}{dx}+13y=0$ .
Given that $y=1$ and $\frac{dy}{dx}=1$ when $x=0$ , find the particular solution.2 marks
Consider the differential equation $\frac{d^2y}{dx^2}-3\frac{dy}{dx}+2y=4x+2$ .
Find a particular integral.3 marks

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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).