Chapter 5: Problem 34
Suppose that you have found a particular solution \(x_{\mathrm{p}}(t)\) of the inhomogeneous equation (5.48) for a driven damped oscillator, so that \(D x_{\mathrm{p}}=f\) in the operator notation of \((5.49) .\) Suppose also that \(x(t)\) is any other solution, so that \(D x=f .\) Prove that the difference \(x-x_{\mathrm{p}}\) must satisfy the corresponding homogeneous equation, \(D\left(x-x_{\mathrm{p}}\right)=0 .\) This is an alternative proof that any solution \(x\) of the inhomogeneous equation can be written as the sum of your particular solution plus a homogeneous solution; that is, \(x=x_{\mathrm{p}}+x_{\mathrm{h}}\).
Short Answer
Step by step solution
Key Concepts
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