Question

Solve$\left(22.9\right)$ to get$(22.10)$ . If needed, see Chapter$\left(8\right)$ , Section 2. The given equation$(D+x)y_0=0\text{toobtain}{y}_0=e^{-\frac{x^2}{2}}\text{.}$

Short answer

Solving,$(D+x)y_0=0$ results $y_0=e^{-\frac{x^2}{2}}$.

Step-by-step solution

Concept of Differential equation

Differential equations are equations that connect one or more derivatives of a function. This implies that their answer is a function.

Step 2:Solve the equations to obtain the solution

Determine equations for the solution as follows:

$\begin{array}{c}(D+x)y_0=0\\ \Rightarrow \frac{d{y}_0}{dx}+x{y}_0=0\\ \Rightarrow \frac{d{y}_0}{d{y}_0}+xdx=0\\ \Rightarrow \log y_0+\frac{x^2}{2}=\log c\end{array}$

So, .$\Rightarrow y_0=c{e}^{\frac{-x^2}{2}}$

Let$y_0\left(0\right)=1$, then,$c=1$.

$y_0=e^{-\frac{x^2}{2}}$

Therefore, solving $(D+x)y_0=0$, results $y_0=e^{-\frac{x^2}{2}}$.