Question

Solve the initial-value problem : (dx)/(dt) +(tant)x=cos^2(t), x(0)=-1

Answer 1

Integrating factor: $\mu(t)=\sec t$

$$\sec t\dfrac{dx}{dt}+\sec t(\tan t)x=\cos t$$

$$\dfrac{d}{dt}(x\sec t)=\cos t$$

$$x\sec t=\int\cos t dt$$

$$x\sec t=\sin t+C$$

$x(0)=-1$

$$-1\cdot\sec(0)=\sin (0)+C$$

$$C=-1$$

$$x\sec t=\sin t-1$$

Or

$$x=\sin t\cos t-\cos t$$