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Question
Solve the following differential equation.
`dy/dx + 2xy = x`
Solution
`dy/dx + 2xy = x`
The given equation is of the form
`dy/dx + py = Q`
where, P = 2x and Q = x
∴ `I.F. = e^(intPdx) = e^ (int ^(2x dx) = e^(x^2)`
∴ Solution of the given equation is
y(I.F.) = `int Q ( I.F.) dx +c`
∴ `y e ^(x^2) int xe^(x^2) dx + c `
In R. H. S., put x2 = t
Differentiating w.r.t. x, we get
2x dx = dt
∴ `ye^(x^2) = int e^t dt/2 + c `
= `1/2 int e^t dt+ c `
= `e^t/2 + c`
∴ `y e ^(x^2) = 1/2 e^(x^2) + c`
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