How many arbitrary constants in the general solution?

Two, both in the complementary function.

CF for complex roots \alpha\pm i\beta?

e^{\alpha x}(A\cos\beta x+B\sin\beta x).

PI trial for f(x)=m\cos\omega x+n\sin\omega x?

\lambda\cos\omega x+\mu\sin\omega x.

What does resonance look like in the solution?

Oscillations whose amplitude grows without bound.

How do you find A and B?

Apply two conditions to the general solution, differentiating once.

Constant coefficient second order equationsEdexcel International A Level Further Maths: Flashcards

Card 1 of 140 of 14 known

Question

General solution of $ay''+by'+cy=f(x)$ is?

Tap or press Space to reveal

Tap card or press Space to flip

See all 14 cards

General solution of $ay''+by'+cy=f(x)$ is?: Complementary function plus particular integral.
How many arbitrary constants in the general solution?: Two, both in the complementary function.
What is the auxiliary equation for $ay''+by'+cy=0$ ?: $am^2+bm+c=0$ .
CF for distinct real roots $m_1,m_2$ ?: $Ae^{m_1x}+Be^{m_2x}$ .
CF for a repeated root $m$ ?: $(A+Bx)e^{mx}$ .
CF for complex roots $\alpha\pm i\beta$ ?: $e^{\alpha x}(A\cos\beta x+B\sin\beta x)$ .
PI trial for $f(x)=ke^{px}$ ?: $\lambda e^{px}$ .
PI trial for $f(x)=A+Bx$ ?: $\lambda+\mu x$ .
PI trial for $f(x)=m\cos\omega x+n\sin\omega x$ ?: $\lambda\cos\omega x+\mu\sin\omega x$ .
PI trial for a quadratic $f(x)$ ?: $\lambda+\mu x+\nu x^2$ .
What if the trial form is in the CF?: Multiply the trial form by $x$ .
PI trial for $y''+4y=\sin2x$ ?: $x(p\cos2x+q\sin2x)$ .
What does resonance look like in the solution?: Oscillations whose amplitude grows without bound.
How do you find $A$ and $B$ ?: Apply two conditions to the general solution, differentiating once.

Exam questions on Constant coefficient second order equations

Consider the differential equation $\frac{d^2y}{dx^2}-5\frac{dy}{dx}+6y=0$ .
Find the particular solution for which $y=1$ and $\frac{dy}{dx}=0$ when $x=0$ .2 marks
Consider the differential equation $\frac{d^2y}{dx^2}+6\frac{dy}{dx}+9y=0$ .
Find the particular solution for which $y=2$ and $\frac{dy}{dx}=0$ when $x=0$ .2 marks
Consider the differential equation $\frac{d^2y}{dx^2}+2\frac{dy}{dx}+5y=5x+7$ .
Find the complementary function.3 marks

See the full worksheet

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).