Answer to Question #215571 in Differential Equations for nasia

Question

solve the initial value problem:

dx/dt+ (tant)x= cos²t , 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

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