Initial-Value Problem: Rewriting Equation for Easier Derivative Calculation

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Homework Statement



((e^t)^2)(dy/dt) + (e^t)^2 (t)(y) = t^3((e^t)^2)

The Attempt at a Solution



Im suppost to figure out how to rewrite the left side of this equation so when you take the derivative of it, it becomes the current left side of the equation as you see it now. we are doing this so we can solve initial value problems however i am stuck. I know the product rule is involved but it was so long ago that i don't remember it.
 
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Is this right? e^(2t) seems to be common.
 
shramana said:
Is this right? e^(2t) seems to be common.

Its these stupid integrating factor method. i swear to god I am doing it right and I am so mad. I am convinced webworks is not working for me. Thankyou for your help
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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