5.2 Solution by Elimination

In this method, unknown functions and their derivatives are successively eliminated until one arrives at a single higher order differential equation containing only one unknown function and its derivatives.

Example 5.2.1.

Solve \(\hspace {0.2cm} x' = -2x + y,\hspace {0.5cm} y' = -4x + 3y + 10\cos t\)

Solution.

From \(1^{\text {st}}\) equation we get \(y = x' + 2x\) substitute in second equation \begin {align*} (x' + 2x)' & = -4x + 3(x' + 2x) + 10\cos t\\\\ x'' + 2x' & = -4x + 3x' + 6x + 10\cos t \end {align*}

\[x'' - x' 2x = 10\cos t\hspace {2cm} *\]

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