Verify Solution of Differential equation

**Aquameatwad** · August 29th 2011, 05:19 PM

Hello, i didn't know whether to post this in the calculus section but since its in my differential equations book I'll start here.

The problem asks to verify the indicated function is a answer of the differential equation.

Verify Solution of Differential equation-1.1-23.jpg

My memory on the natural e is rusty. Any guidance? I feel like this should be easier than it looks.

Thanks

**skeeter** · August 29th 2011, 05:45 PM

$y = e^{-x^2}\left[\int_0^x e^{t^2} \, dt + c_1\right]$

product rule to find $\frac{dy}{dx}$ ...

$\frac{dy}{dx} = e^{-x^2} \cdot e^{x^2} - 2xe^{-x^2}\left[\int_0^x e^{t^2} \, dt + c_1\right]$

$\frac{dy}{dx} = 1 - 2xy$

$(1 - 2xy) + 2xy = 1$

**Aquameatwad** · August 29th 2011, 07:51 PM

Do you have to solve the integral? i used mathway for guidance and it gave me zero. Heres a link

Mathway: Evaluate the Integral

**Prove It** · August 29th 2011, 08:49 PM

Originally Posted by skeeter

$y = e^{-x^2}\left[\int_0^x e^{t^2} \, dt + c_1\right]$

product rule to find $\frac{dy}{dx}$ ...

$\frac{dy}{dx} = e^{-x^2} \cdot e^{x^2} - 2xe^{-x^2}\left[\int_0^x e^{t^2} \, dt + c_1\right]$

$\frac{dy}{dx} = 1 - 2xy$

$(1 - 2xy) + 2xy = 1$

Since it's first order linear, you could solve the DE using the Integrating Factor.

$\displaystyle \frac{dy}{dx} + 2x\,y = 1$ , the integrating factor is $\displaystyle e^{\int{2x\,dx}} = e^{x^2}$ , so multiplying both sides of the DE by the integrating factor gives

$\displaystyle \begin{align*}e^{x^2}\frac{dy}{dx} + 2x\,e^{x^2}y &= e^{x^2} \\ \frac{d}{dx}\left(e^{x^2}y\right) &= e^{x^2} \\ e^{x^2}y &= \int{e^{x^2}\,dx} + C_1 \\ y &= e^{-x^2}\int{e^{x^2}\,dx} + C_1e^{-x^2} \end{align*}$

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