5.2 Solution by Elimination
In this method, unknown functions and their derivatives are successively eliminated until a single higher-order differential equation containing only one unknown function remains.
Solution. The first equation gives \(y = x' + 2x\). Substituting into the second, \[(x' + 2x)' = -4x + 3(x' + 2x) + 10\cos t,\qquad \text {that is}\qquad x'' + 2x' = -4x + 3x' + 6x + 10\cos t ,\] so that \begin {equation} x'' - x' - 2x = 10\cos t , \label {eq:elim} \end {equation} a single second-order equation for \(x\), solved in the next subsection.
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