How to Verify Solutions to Second Order Differential Equations

If you have a function and want to show that it is a solution to a differential equation, you must insert it into the equation and check if the left and the right-hand side are equal. This involves finding both the first and the second derivatives of the given function, so that you can insert them into the equation.

Example 1

Check if $y_{0} = x \cdot e^{x}$ is a solution to the differential equation

y^{″} - 2 y^{'} + y = 0

Find the derivative of $y_{0}$ :

y_{0}^{'} = e^{x} + x \cdot e^{x}

Find the second derivative of $y_{0}$ :

y_{0}^{″} = 2 e^{x} + x e^{x}

Enter the values for $y_{0}$ , $y_{0}^{'}$ and $y_{0}^{″}$ into the differential equation:

\begin{array}{l} L. H. S & = y^{″} - 2 y^{'} + y \\ = 2 e^{x} + x e^{x} \\ - 2 (e^{x} + x \cdot e^{x}) + x \cdot e^{x} \\ = 0 \\ = R.H.S \end{array}

\begin{array}{l} L. H. S & = y^{″} - 2 y^{'} + y \\ = 2 e^{x} + x e^{x} - 2 (e^{x} + x \cdot e^{x}) + x \cdot e^{x} \\ = 0 \\ = R.H.S \end{array}

As the left and the right-hand side are equal, you have shown that the function is a solution to the differential equation.

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