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 . 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.

role="math" localid="1663829534574" $\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathit{\pi }\mathbf{\right)}\mathbf{=}\mathbf{5}$

Short answer

The explicit solution of the differential equation is $y=\cos 2x+\frac{5}{2}\sin 2x$.

Step-by-step solution

First boundary condition

Differentiating the family of solutions given us the first derivative,

$y\text{'}=-2c_1\sin 2x+2c_2\cos 2x$

Substituting the first boundary condition $y\left(0\right)=1$to the given two parameter family of the solution then gives,

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

Second boundary condition

Substituting the secondary boundary condition $y\text{'}\left(\pi \right)=5$to the given two parameter family of solution gives,

$$\begin{gathered} y\text{'}\left(\pi \right)=-2c_1\sin \left(2\right(\pi \left)\right)+2c_2\cos \left(2\right(\pi \left)\right) \\ 5=-2c_1\left(0\right)+2c_2\left(1\right) \\ 5=2c_2 \\ {c}_2=\frac{5}{2} \end{gathered}$$

Hence, the constants have values,

$c_1=1$and $c_2=\frac{5}{2}$.

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$y=\cos 2x+\frac{5}{2}\sin 2x$