Q3 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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

$\mathbf{y}\mathbf{=}\mathbf{sin} \mathbf{x}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}$, $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{{\mathbf{dx}}^{\mathbf{2}}}\mathbf{+}\mathbf{y}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}$

Step-by-Step Solution

Answer

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

Step 1: Differentiating the given equation w.r.t. (with respect to) x

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

$\frac{\mathrm{dy}}{\mathrm{dx}}=\cos \mathrm{x}+2\mathrm{x}$

Again, differentiating with respect to x,

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

Step 2: Simplification

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

$$\begin{gathered} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+\mathrm{y}=-\sin \mathrm{x}+2+\sin \mathrm{x}+{\mathrm{x}}^2 \\ \frac{{\displaystyle {\mathrm{d}}^2\mathrm{y}}}{{\displaystyle {\mathrm{dx}}^2}}+\mathrm{y}={\mathrm{x}}^2+2 \end{gathered}$$

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

Hence, $\mathbf{y}\mathbf{=}\mathbf{sinx}\mathbf{+}{\mathbf{x}}^{\mathbf{2}}$ is a solution to the differential equation $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{y}}{{\mathbf{dx}}^{\mathbf{2}}}\mathbf{+}\mathbf{y}\mathbf{=}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}$.