Second Order Equations (AQA A-Level Further Maths): Quizzes

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Second Order Equations
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What defines a second order differential equation?

Contains d2ydx2\frac{d^2y}{dx^2}, no higher derivatives

What makes an equation homogeneous?

The right-hand side equals zero: f(x)=0f(x) = 0

For auxiliary equation (m+3)(m−2)=0(m+3)(m-2) = 0, what is the general solution?

y=Ae−3x+Be2xy = Ae^{-3x} + Be^{2x}

For repeated root m=4m = 4, what is the general solution form?

y=(Ax+B)e4xy = (Ax + B)e^{4x}

For complex roots m=2±3im = 2 \pm 3i, what is the general solution?

y=e2x(Acos⁡3x+Bsin⁡3x)y = e^{2x}(A\cos 3x + B\sin 3x)

What is the general solution formula?

GS = CF + PI

For f(x)=12x−8f(x) = 12x - 8, what form should the PI take?

y=ax+by = ax + b

If CF is Ae2x+Be3xAe^{2x} + Be^{3x} and f(x)=6e2xf(x) = 6e^{2x}, what PI form to try?

y=axe2xy = axe^{2x}

What are boundary conditions used for?

Find specific values of constants AA and BB

For auxiliary equation m2−8m+16=0m^2 - 8m + 16 = 0, what type of roots?

Repeated root

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