Question

by variation of parameters
y''+y=sec(x)

Answer 1

So general solution= yc+yp. Yc is the solution to the homogenous DE which is fairly simple to solve. yc= c1cost+c2sint

Now, to get Yp, you use this: yp= v1y1+v2y2 where y1=cost and y2=sint which means you need to solve v1 and v2. So to solve for those, you use these two equations: v1'y1+v2'y2=0 and v1'y1'+v2'y2'=g(t) where g(t) in this case is secx. Then you need to eliminate a variable to solve this system and integrate v1' to find v1. Then solve for v2' and integrate that to solve for v2. Then solve for yp and combine with yc.