Question

Solve the following differential equation:

`dy/dx + y/x = x^3 - 3`

Answer 1

`dy/dx + y/x = x^3 - 3`   ...(1)

This is the linear differential equation of the form

`dy/dx + P * y = Q`, where `P = 1/x` and Q = x³ – 3

∴ I.F. = `e^(int Pdx)` 

= `e^(int 1/x dx)`

= `e^(log x)` 

= x

∴ The solution of (1) is given by

y(I.F.) = ∫ Q. (I.F.) dx + c₁

∴ `y * x = int (x^3 - 3)x  dx + c_1`
∴ `xy = int (x^4 - 3x) dx + c_1`

∴ `xy = x^5/5 - 3 * x^2/2 + c_1`

∴ `x^5/5 - (3x^2)/2 - xy = c`, where c = – c₁

∴ This is the general solution.