Numerical Methods in Context (Edexcel A-Level Mathematics): Quizzes

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Numerical Methods in Context
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What is the general form of a non-homogeneous second-order differential equation?

ad2ydx2+bdydx+cy=F(x)a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + cy = F(x)

For RHS 13sin⁡(3x)13 \sin(3x), what form should be assumed for the particular solution?

λsin⁡(3x)+μcos⁡(3x)\lambda \sin(3x) + \mu \cos(3x)

What is the characteristic equation for d2ydx2−5dydx+6y=0\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = 0?

m2−5m+6=0m^2 - 5m + 6 = 0

What are the roots when factorizing (m−2)(m−3)=0(m - 2)(m - 3) = 0?

m=2m = 2 and m=3m = 3

With roots m=2m = 2 and m=3m = 3, what is the complementary function?

Ae2x+Be3xAe^{2x} + Be^{3x}

Is the complementary function alone a solution to the non-homogeneous equation?

No, it equals zero, not F(x)F(x)

For RHS 3x23x^2, what form should the particular integral take?

ax2+bx+cax^2 + bx + c

For RHS λekx\lambda e^{kx}, what form should the particular integral take?

aekxae^{kx}

What is the complete solution to a non-homogeneous SODE?

Particular integral + Complementary function

If e2xe^{2x} is in C.F. and RHS is e2xe^{2x}, what form should P.I. take?

λxe2x\lambda x e^{2x}

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