Question

(a) Use a CAS to graph the solution curve of the initial-value

problem in Problem$\frac{dy}{dx}-2xy=-1,y\left(0\right)=\sqrt{\pi /2}$on the interval$\left(-\infty ,\infty \right)$

(b) Use tables or a CAS to value the value

Short answer

$y\left(2\right)\approx 0.22633$

Step-by-step solution

(a) Given Information.

The given curve is:

$erfc\left(x\right)=\frac{2}{\sqrt{\pi }}\underset{x}{\overset{\infty }{\int }}{e}^{-t^2}dt$

Indicating value in graph

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

(b) Appling CAS

In $y\left(x\right)$, insert the value$x=2$

$$\begin{gathered} y=\frac{\sqrt{\pi }}{2}{e}^{x^2}erfc\left(x\right) \\ y\left(2\right)=\frac{\sqrt{\pi }}{2}{e}^{2^2}erfc\left(2\right) \\ y\left(2\right)=\frac{\sqrt{\pi }}{2}{e}^{4}erfc\left(2\right) \\ y\left(2\right)\approx 0.22633 \end{gathered}$$

$$\begin{gathered} y\left(2\right)=\frac{\sqrt{\pi }}{2}{e}^{2^2}erfc\left(2\right) \\ y\left(2\right)=\frac{\sqrt{\pi }}{2}{e}^{4}erfc\left(2\right) \\ y\left(2\right)\approx 0.22633 \end{gathered}$$

Result

So, the result is$y\left(2\right)\approx 0.22633$