Question

In a differential equation, y''+y=0, why do we automatically assume Acosx + Bsinx is the solution?
Is it just because we solved it once, so we don't have to worry about it again?
But what if we throw another term into the equation like y''+y'+y=0?

Answer 1

You need to solve for roots

Answer 2

If the roots are distinct general solution is \[y= c1*e^{(r1*x)} + c2*e^{(r2*x)}\]

Answer 3

complex \[y= A \cos(Bx) + A2 \sin(Bx) ) e^{(ax)} , for a+Bi\]

Answer 4

Oh! This is constant coefficients so our characteristic equation is r^2 +1 = 0
and we have complex roots so then we know we can use cos(x) and sin(x)!

Thanks!!