Nonhomogeneous equations

Complementary plus particular solution, undetermined coefficients for the usual forcing shapes, and variation of parameters for when guessing a form fails.

Notes on Diffy Qs: Differential Equations for Engineers, Nonhomogeneous equations

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6 exercises

Exercise 101

Find a particular solution to $y''-y'+y=2\sin(3x)$ .

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Exercise 102

Find a particular solution to $y''+2y=e^x + x^3$ .
Find the general solution.

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Exercise 103

Solve $y''+2y'+y = x^2$ , $y(0)=1$ , $y'(0)=2$ .

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Exercise 104

Use variation of parameters to find a particular solution of $y''-y = \frac{1}{e^x+e^{-x}}$ .

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Exercise 105

For an arbitrary constant $c$ find the general solution to $y''-2y=\sin(x+c)$ .

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Exercise 106

Undetermined coefficients can sometimes be used to guess a particular solution to other equations than those with constant coefficients. Find a polynomial $y(x)$ that solves $y'+ x y = x^3+2x^2+5x+2$ .

Note: Not every right-hand side will allow a polynomial solution, for example, $y'+xy=1$ does not, but a technique based on undetermined coefficients does work, see the “Power series methods” chapter.

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