Solve the following ODE using integrating factors 12yx 2xyx

Solve the following ODE using integrating factors. 1/2y\'(x) + 2/xy(x) = 1/x^4 (with x notequalto 0)

Solution

Multiplying the eqn by 2,

y\'(x) + (4/x) y(x) = 2/x^4...............eqn1

Let u(x) = e^(Integral (4/x) dx) = x^4

Multiplying both sides of eqn1 by x^4 we get

(x^4) y\'(x) + (4x^3) y(x) = 2

d/dx [x^4 * y(x)] = 2

Integrating both the sides with respect to x we get,

x^4 * y(x) = 2x + C

y(x) = (2 / x^3) + (C / x^4)