What are the steps for solving a 2nd order differential equation?

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Firstly, suppose y1 is a solution to y''+ay'+by=g(x). What then is a solution of y''+ay'+by=kg(x), where k is an arbitrary non-zero constant?

Secondly, note that a general solution to a non-homogenous linear ODE is its homogenous solution plus one particular solution to its nonhomogenous ODE.
 
solution to y''+ay'+by=kg(x) is ky1?

so the solution to y''+ay'+by=3g(x) is the complimentyery function + particular integral = 3y1 + y2?
 
That should be correct.
 
thanks!
http://img237.imageshack.us/img237/3676/167uqnbsi6.gif
 
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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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