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Question
Solve the following differential equation:
`dy/dx + y/x = x^3 - 3`
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Solution
`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.
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