Question

Solve the following differential equations by method (a) or (b) above.${\mathit{y}}^{\mathbf{\text{'}\text{'}}}\mathbf{+}\mathbf{2}\mathit{x}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{0}$

. Hint: The solution is$\mathit{y}\mathbf{=}{\mathit{c}}_{\mathbf{1}}\mathbf{\text{erf}}\mathit{x}\mathbf{+}{\mathit{c}}_{\mathbf{2}}$see Chapter 11, Section 9 for the definition of $\mathbf{\text{erf}}\mathit{x}$.

Short answer

The general solution of$y^{\text{'}\text{'}}+2x{y}^{\text{'}}=0$ is.$y=c_1\text{erf}\left(x\right)+c_2$

Step-by-step solution

Given Information. 

The given equation is $y^{\text{'}\text{'}}+2x{y}^{\text{'}}=0$.

Definition/ Concept.

The formula for .$\text{erf}\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt$

Solve the differential equation.

Using the method (a).

Let $y^{\text{'}}=p$and .$y^{\text{'}\text{'}}=p^{\text{'}}$

Now put these values in the given equation as:

$\begin{array}{c}{y}^{\text{'}\text{'}}+2x{y}^{\text{'}}=0\\ p^{\text{'}}+2xp=0\\ \frac{dp}{dx}+2xp=0\end{array}$

Now separating variables as:

$\begin{array}{c}\frac{dp}{dx}+2xp=0\\ \frac{dp}{dx}=-2xp\\ \frac{dp}{p}=-2xdx\end{array}$

Now integrating both sides as:

$\begin{array}{c}\int \frac{dp}{p}=-2\int xdx\Rightarrow p=e^{-x^2+c}\\ \ln p=-2\cdot \frac{x^2}{2}+c\Rightarrow p=e^{-x^2}\cdot e^c\\ \ln p=-x^2+c\Rightarrow p=A{e}^{-x^2}\end{array}$

Where .$A=e^c$

Now put $p=y^{\text{'}}$again $p=c_1{e}^{-x^2}$in as:

$\begin{array}{c}p=c_1{e}^{-x^2}\\ y^{\text{'}}=A{e}^{-x^2}\\ \frac{dy}{dx}=A{e}^{-x^2}\\ dy=A{e}^{-x^2}dx\end{array}$

Integrating both sides as:

$\begin{array}{c}dy=c_1{e}^{-x^2}dx\\ \int dy=A\int e^{-x^2}dx\end{array}$

Now using.$\int e^{-x^2}dx=\frac{1}{2}\sqrt{\pi }\text{erf}\left(x\right)$

So, integrating as:

$\begin{array}{c}\int dy=A\int e^{-x^2}dx\\ y=A\times \frac{1}{2}\sqrt{\pi }\text{erf}\left(x\right)+c_2\\ y=c_1\text{erf}\left(x\right)+c_2\end{array}$

Where.$c_1=\frac{A}{2}\sqrt{\pi }$

Hence, the general solution of $y^{\text{'}\text{'}}+2x{y}^{\text{'}}=0$is.$y=c_1\text{erf}\left(x\right)+c_2$