Solve x2 y′′ + 3x y′ + y = 1(1−x)2 .

This is a Cauchy-Euler differential equation. The general form of the solution is y = x^r. We can solve it step by step:

1. First, we substitute y = x^r into the differential equation. The derivatives are y' = rx^(r-1) and y'' = r(r-1)*x^(r-2).

2. Substituting these into the equation gives us: x^2 * r*(r-1)x^(r-2) + 3x * rx^(r-1) + x^r = x^r * (1 - x)^2.

3. Simplifying this gives us: r^2 - 2r + 1 = (1 - x)^2.

4. This is a quadratic equation in r. We can solve it using the quadratic formula: r = [2 ± sqrt(4 - 411)] / (2*1) = 1.

5. Therefore, the general solution of the differential equation is y = x^r = x^1 = x.

6. We can check this solution by substitifying it back into the original differential equation.