A solution to a homogeneous system

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Homework Statement



I know there is a polynomial that is a solution to the equation:

3/2f(x) - x/2f'(x) - f''(x)=x



Homework Equations





The Attempt at a Solution


I tried many polynomials of degrees 1,2 and 3, but they do not work in my equation
 
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What happens when you back-substitute f(x)=x?
 
I think you want http://en.wikipedia.org/wiki/Method_of_variation_of_parameters" for this problem.
 
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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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