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.

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

Short answer

No solution

Step-by-step solution

First boundary condition

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

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

Second boundary condition

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

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

Hence, we only one constant value $c_1=0$and we cannot find the exact value of $c_2$using the given boundary conditions.

Therefore, it is not possible to have a solution of the differential equation that satisfies the given side conditions.