Question

Proceed as in Example 7 and express the solution of the given initial value problem in terms of erf (x) (Problem 43) and erfc (Problem 44).

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{-}\mathbf{2}\mathbf{xy}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{1}$

Short answer

So, the solution of the given initial value is $y=\frac{\sqrt{p}}{2}{e}^{x^2}\left[erf\right(x)-erf(1\left)\right]+e^{x^2-1}$.

Step-by-step solution

Form of linear equation

The linear differential equation is of the form $\frac{\mathbf{dy}}{\mathbf{dx}}+\mathbf{Py}=\mathbf{Q}$, where P and Q are numeric constants or functions in x.

Evaluation

Linear differential equation of the first order

$y^{\text{'}}+P\left(x\right)y=f\left(x\right)$

The integrating factor is given by

$e^{\int P\left(x\right)dx}$

and the error function is given by

$\begin{array}{l}erf\left(x\right)=\frac{2}{\sqrt{p}}{\int }_0^x{e}^{-t^2}dt\\ {\int }_0^x{e}^{-t^2}dt=\frac{\sqrt{p}}{2}erf\left(x\right)\end{array}$

Based on (1) and the given differential equation, we obtain

role="math" $P\left(x\right)=-2x\mathrm{and}f\left(x\right)=1$

Finding integrated factor

Use the formula (2), to get integrating factor

$\begin{array}{l}{e}^{\int P\left(x\right)dx}=e^{-2\int xdx}\\ =e^{-x^2}\end{array}$

Thus, the solution is $y{e}^{-x^2}={\int }_0^x{e}^{-t^2}\mathrm{d}t+c$.

(3) $\frac{\sqrt{p}}{2}erf\left(x\right)+c$

Therefore,

$y=\frac{\sqrt{\pi }}{2}erf\left(x\right)e^{x^2}+c{e}^{x^2}$

Use the initial condition y(1) = 1 to get

$\begin{array}{l}y=\frac{\sqrt{\pi }}{2}erf\left(x\right)e^{x^2}+c{e}^{x^2}\\ 1=\frac{\sqrt{\pi }}{2}erf\left(1\right)e+ce\\ c=e^{-1}-\frac{\sqrt{\pi }}{2}erf\left(1\right)e\end{array}$

Substitution

Substitute the value $c=e^{-1}-\frac{\sqrt{\pi }}{2}erf\left(1\right)e$in the equation (4).

$\begin{array}{l}y=\frac{\sqrt{p}}{2}erf\left(x\right)e^{x^2}+\left(e^{-1}-\frac{\sqrt{p}}{2}erf\left(1\right)e\right)e^{x^2}\\ y=\frac{\sqrt{p}}{2}{e}^{x^2}\left[erf\right(x)-erf(1\left)\right]+e^{x^2-1}\end{array}$

So, the required solution is $y=\frac{\sqrt{p}}{2}{e}^{x^2}\left[erf\right(x)-erf(1\left)\right]+e^{x^2-1}$.