Question

Solve the following differential equation by using separation of variables method: dp/dt =(1+p^2)cos(t)/psin(t)

Answer 1

$\frac{dp}{dt}=\frac{(1+p^2)cost}{psint}\\\text{Cross multiplying, we have,}\\dp(psint)=(1+p^2)costdt\\\implies\frac{p}{1+p^2}dp=\frac{cost}{sint}dt\\\text{Integrating both sides, we have }\\\frac{1}{2}\ln(1+p^2)=\ln sint+\ln c\\\text{Taking the exponential of both sides, we have}\\(1+p^2)^{\frac{1}{2}}=csint$