Solve the following differential equation x^2 dy/dx + y = 3x^4 e^1/x , y(1)=1Help me please, thank you.

Question

Solve the following differential equation x^2 dy/dx + y = 3x^4 e^1/x , y(1)=1. Help me please, thank you.

Answer 1

x²dy/dx + y = 3x⁴e^1/x; dy/dx + 1/x²y = 3x²e^1/x;

∫Integrated factor M = e^{∫1/x^2dx} = e^-1/x.

Multiplaying we get: e^-1/xdy/dx + 1/x²e^-1/xy = 3x²;

d/dx(ye^-1/x) = 3x²; ye^-1/x = x³ + C; y = (x³ + C)·e^1/x; 1 = (1 + C)e; C = 1/e - 1; y = (x³ + 1/e - 1)e^1/x