help find the general solution of the following first order differential equation.

Question

t*(dy/dt) = (-y/t) + ((exp(1/t)/t^3)

could you show steps in working to sol.

Answer 1

I like x instead of t. You have

xy' = -y/x + e^(1/x) / x^3

Change to the form

y' + yP(x) = Q(x)

y' + y*1/x² = e^1/x/x⁴

Now find the integrating factor

IF = exp(Int(P(x)) = exp(Int(1/x²)) = e^-1/x

Now, y = 1/e^-1/x Int(Q(x)*IF)
= e^1/x Int(e^1/x/x⁴ * e^-1/x dx)
= e^1/x*(-1/3x³ + C)

y = Ce^1/x - e^1/x/3x³