Differential equations

Question

Find the general solution of the given differential equation.

xdy/dx+ 3y = x3 − x

Answer 1

From the form of the RHS of the equation, try a polynomial:

y(x) = ax^3 + bx^2 + cx + d

as a particular solution. Then

xy'(x) + 3y(x) = 6ax^3 + 5bx^2 + 4cx + 3d

Choosing a = 1/6, b =0, c = -1/4 and d=0 gives us the particular solution we need. To obtain the general solution, consider the homogeneous equation

xy' + 3y = 0.

Separating variables and solving dy/y = -3 dx /x gives y = A/x^3 as a solution to the homogeneous equation for an arbitrary constant A. Thus our solution is

y = Ax^{-3} + (x^3)/6 - x/4

as a general solution to this equation.