Solving system of diff equations using elimination

Question

I am trying to solve this system of differential equations via elimination method but I am stuck. Any help on arriving at correct solution.

Answer 1

y₁'' = y₂^'; y₁'' + y₁ = 1/cosx; r² + 1 = 0; r = ±i; y₁₁ = cosx; y₁₂ = sinx; W = 1

y_1P = - cosx∫sinx·1/cosxdx + sinx∫cosx·1/cosxdx = cosxlnIcosxI + xsinx

y₁ = Acosx + Bsinx + cosxlnIcosxI + xsinx;

y₂ = - Asinx + Bcosx - sinxlnIcosxI - sinx + sinx + xcosx = - Asinx + Bcosx - sinxlnIcosxI + xcosx