Answer to Question #177758 in Differential Equations for Zain Ul Abideen Khan

Question

Solve the following non-homogeneous linear ODE of first order

dy/dx + 3y/x =6x^2

Answer 1

Solution

Let’s find solution of the homogeneous equation dy₀/dx + 3y₀/x = 0 in the form y₀ = Cx^k. Substitution into this equation gives

Ckx^k-1 + 3Cx^k-1 = 0  => k+3 = 0 => k = -3. = y₀ = Cx^-3.

Partial solution of nonhomogeneous equation let’s find as the expression y₁ = Ax³. Substitution into equation gives

3Ax² + 3Ax² = 6x² =>  A = 1 => y₁ = x³.

Therefor the solution of the given equation is

y = y₀+y₁ = Cx^-3+ x³

Answer

y = Cx^-3+ x³