Question

Question: In Problems 37-40 solve the given initial-value problem.

$$\begin{gathered} \mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathit{y}\mathbf{=}\mathbf{0} \\ \mathit{y}\mathbf{\text{'}}\left(0\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathit{y}\mathbf{\text{'}}\left(\frac{\pi }{2}\right)\mathbf{=}\mathbf{0} \end{gathered}$$

Short answer

$y=0$

Step-by-step solution

Determine the initial value of differential equation

The second order differential equation:

$y\text{'}\text{'}+y=0$

Initial conditions:

$y\text{'}\left(0\right)=0,y\text{'}\left(\frac{\pi }{2}\right)=0$

Consider$y=e^{mx}$as the solution of the differential equation,

Substitute$y=e^{mx},y\text{'}=m{e}^{mx}$ and$y\text{'}\text{'}=m^2{e}^{mx}$ into$y\text{'}\text{'}+y=0$ to obtain the auxiliary equation.

$$\begin{gathered} m^2{e}^{mx}+e^{mx}=0 \\ {e}^{mx}=\left(m^2+1\right)=0 \end{gathered}$$

For any real$x,e^{mx}\ne 0$, we have

$$\begin{gathered} m^2+1=0 \\ {m}^2=-1 \end{gathered}$$

Roots are:

data-custom-editor="chemistry" $m_{1,2}=\pm i$

Find the general solution of the homogeneous equation

conjugate and complex.

$\alpha =0 and \beta =1$the general solution of this homogeneous equation can be given as$y\text{'}\text{'}+y=0$

$y=c_1\cos x+c_2\sin x······\left(1\right)$

Arbitrary constants:$c_1 \text{and} c_2$

Fourth, now to apply the given conditions in the general solution of our given differential equations as the following technique:

First derivative for the general solution (1) as

Given differential

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

Apply with the point$\left(x,y\right)=\left(0,0\right)$ in the general solution as

$$\begin{gathered} 0=c_1\sin \left(0\right)+c_2\cos \left(0\right) \\ {c}_2=0 \end{gathered}$$

Then apply with point $\left(x,y\text{'}\right)=\left(\frac{\pi }{2},0\right)$

$$\begin{gathered} 0=-c_1\sin \left(\frac{\pi }{2}\right)+c_2\cos \left(\frac{\pi }{2}\right) \\ {c}_1=0 \end{gathered}$$

substitute with the values of$c_1 \text{and} c_2$ equation into the general solution of our given differential equation.

y = 0

Hence the general solution is y = 0.