Diff EQ Intro - Verify Family of Functions as Solution

[jgiarrusso]

Homework Statement

Verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution.

Homework Equations

dy/dx + 2xy = 1; y = e^-x2$$\int$$(from 0 to x)e^t2dt + c₁e^-x2

The Attempt at a Solution

I have only had one class period in differential so far and we didn't get to go over much material. I imagine that one would need to differentiate y(x) with respect to x and plug into the first equation. However, I'm not quite sure what to do with the integral with respect to t. I tried to integrate it, and got e^t2/(2t), but evaluating that at 0 would cause an implosion. If I differentiate with respect to x, I don't think I can just treat it as a constant because it's evaluated from 0 to x. Could I please get a nudge in the right direction?

 

[Raskolnikov]

I'm not quite sure what to do with the integral with respect to t.

This is just simple application of Fundamental Theorem of Calculus.

If $$F(x) = \int_a^x f(t) dt$$ then $$F'(x) = f(x),$$ given of course that f(x) is continuous on [a, x].

 

[jgiarrusso]

So then dy/dx would be:

dy/dx = e^-x2 * e^x2 - 2xe^-x2$$\int$$(from 0 to x)e^t2dt - 2xc₁e^-x2

And then plugging it into the differential equation, it all cancels out. Thank you so much!