Is reduction of order a simpler method for solving this integral?

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The problem has ty'' - (1 + t)y' + y = (t^2)e^2t

y1 = 1 + t

Solve by reduction of order

When I solve by variation of parameters I get:

y = .5te^2t - .5e^2t + ce^t + d(1 + t)

But solving with reduction of order gives very difficult integrals
 

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Well I should have seen this one

(ue^u - e^u)/u^2 is a quotient rule.
 

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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