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- #1

- Feb 5, 2012

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One of my friends gave me the following question. I am posting the question and the answer here so that he could check his work.

**Question:**

This question concerns the differential equation,

\[x\frac{d^{2}y}{dx^2}-(x+1)\frac{dy}{dx}+y=x^2\]

and the associated homogeneous differential equation,Wronskian - Wikipedia, the free encyclopedia

\[x\frac{d^{2}y}{dx^2}-(x+1)\frac{dy}{dx}+y=0\]

a) Show that \(y_{1}(x)=e^x\) is a solution of the homogeneous differential equation.

b) Use the method of reduction of order to show that a second linearly independent solution of the homogeneous differential equation is, \(y_{2}(x)=x+1\).

**Hint:**

\(\int xe^{-x}\,dx=-(x+1)e^{-x}+C\)

c) Use the method of variation of parameters to find the general solution of the given non-homogeneous differential equation.

**Hint:**

Write the differential equation in standard form and remember the hint from part (b).