Question

Solve the differential equation x²d²y/ dx² -2x(1 + x)dy/dx + 2(1 + x)y = x3​

Answer 1

Answer:

−2x(1+x)dydx+2(1+x)y

admit a solution y1=x to the associated homogeneous equation. This will be the case if the difference between the coefficients of y’ and y is a factor of x and opposite signs. So with one solution, you can obtain a fundamental solution set via reduction of order. Letting y2=y1u(x) and applying reduction of order, you get a second solution to the homogeneous equation y2=xe2x . With this, you can find a particular solution using variation of parameters. Put yp=u1y1+u2y2 . There are simple formulas for the u′s in terms of y1,y2 , the right hand side (which here is x ) and the Wronskian of y1 and y2 . That variation of parameters process produces yp=−12x2−14x . The latter term can be combined with the complementary solution to give the solution

y=c1x+c2xe2x−12x2.

If you look for a series solution, you’ll likely come across that first linear solution and can obtain the rest using reduction of order and variation of parameters.

Step-by-step explanation:

hope it will help you