Question

In Problems 39–44, $\mathit{y}\mathbf{=}{\mathit{c}}_{\mathbf{1}}\mathit{c}\mathit{o}\mathit{s}\mathbf{2}\mathit{x}\mathbf{+}{\mathit{c}}_{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathbf{2}\mathit{x}$is a two-parameter family of solutions of the second-order DE $\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{=}\mathbf{0}$. If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.

$\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathit{y}\mathbf{(}\mathit{\pi }\mathbf{/}\mathbf{4}\mathbf{)}\mathbf{=}\mathbf{3}$

Short answer

The explicit solution of the differential equation is $y=3\sin 2x$

Step-by-step solution

First boundary condition

Substituting the first boundary condition to the given two parameter family of solution gives,

$$\begin{gathered} \left(0\right)=c_1\cos \left(2\right(0\left)\right)+c_2\sin \left(2\right(0\left)\right) \\ 0=c_1\cos 0+c_2\sin 0 \\ 0=c_1\left(1\right)+c_2\left(0\right) \\ 0=c_1 \end{gathered}$$

Second boundary condition

Substituting the secondary boundary condition to the given two parameter family of solution gives,

$$\begin{gathered} y\left(\frac{\pi }{4}\right)=c_1\cos \left(2\right(\frac{\pi }{4}\left)\right)+c_2\sin \left(2\right(\frac{\pi }{4}\left)\right) \\ 3=c_1\cos \left(\frac{\pi }{2}\right)+c_2\sin \left(\frac{\pi }{2}\right) \\ 3=c_1\left(0\right)+c_2\left(1\right) \\ 3=c_2 \end{gathered}$$

Hence, the constants have values,

$c_1=0$and $c_2=3$.

Therefore, substituting values of $c_1$and $c_2$into the family of solutions gives us the explicit solution of the differential equation.

Hence the solution is$y=3\sin 2x$