Solve the initial value problem given below dxdt xt t 0 x2

Solve the initial value problem given below dx/dt + x/t -t = 0, x(2) = 1/3

dx/dt + x/t = t, x(2) = 1/3

Integrating Factor can be calculated by using the method

I.F. = e^Integral{p(t)dt} = e^Integral{1/t*dt} = e^{ln(t}} = t

Hence multiply the equation by integrating factor we get

tdx/dt + x = t^2

d/dt [xt] = t^2

Taking integral on both sides

xt = Integral(t^2 dt)

xt = 1/3 * t^3 + C

x = 1/3 * t^2 + Ct^{-1}

Since x(2) = 1/3, we get

1/3 = 4/3 + C/2

C = -2

Hence, Final answer is

x = 1/3 * t^2 - 2t^{-1}