Question

Question: In Problems 3–8, determine whether the given function is a solution to the given differential equation.

$\mathbf{y}\mathbf{=}\mathbf{3}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{2}\mathbf{x}\mathbf{+}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}$, $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{5}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}$

Short answer

The given function is a solution to the given differential equation.

Step-by-step solution

Differentiating the given equation w.r.t. (with respect to) x

Firstly, we will differentiate $\mathrm{y}=3\sin 2\mathrm{x}+{\mathrm{e}}^{-\mathrm{x}}$with respect to x,

$\frac{\mathrm{dy}}{\mathrm{dx}}=6\cos 2\mathrm{x}-{\mathrm{e}}^{-\mathrm{x}}$

Again, differentiatingthe given function with respect to x,

$\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}=-12\sin 2\mathrm{x}+{\mathrm{e}}^{-\mathrm{x}}$

Simplification

Putting the values from step 1 in the L.H.S. (Left-hand side) of the given differential equation,

$$\begin{gathered} \mathrm{y}\text{'}\text{'}+4\mathrm{y}=-12\sin 2\mathrm{x}+{\mathrm{e}}^{-\mathrm{x}}+4\left(3\sin 2\mathrm{x}+{\mathrm{e}}^{-\mathrm{x}}\right) \\ \mathrm{y}\text{'}\text{'}+4\mathrm{y}=-12\sin 2\mathrm{x}+{\mathrm{e}}^{-\mathrm{x}}+12\sin 2\mathrm{x}+4{\mathrm{e}}^{-\mathrm{x}} \\ \mathrm{y}\text{'}\text{'}+4\mathrm{y}=5{\mathrm{e}}^{-\mathrm{x}} \end{gathered}$$

which is the same as the R.HS. (Right-hand side) of the given differential equation.

Hence, $\mathbf{y}\mathbf{=}\mathbf{3}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{2}\mathbf{x}\mathbf{+}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}$is a solution to the differential equation $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{5}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}$.