Question

Find the solution of the differential equation that satisfies the given initial condition. (du)/(dt) = (2t + sec^2 t)/(2u), u(0) = -5

Answer 1

This ODE is separable:

$\dfrac{\mathrm du}{\mathrm dt}=\dfrac{2t+\sec^2t}{2u}$

$\implies2u\,\mathrm du=(2t+\sec^2t)\,\mathrm dt$

Integrating both sides gives

$u^2=t^2+\tan t+C$

Given that $u(0)=-5$, we have

$(-5)^2=0^2+\tan0+C\implies C=25$

so that the particular solution to the IVP is

$u(t)^2=t^2+\tan t+25$

$\boxed{u(t)=\sqrt{t^2+\tan t+25}}$