Let the roots of the auxiliary equation r2 a1r a2 0 be a ( i From Problem 25c, it follows, just as in the real case, that y c1e(a( i)x c2e(a i)x satisfies (D2 a1D a2)y 0 Show that this solution can be rewritten in the form Y C1eax cos x C2eax sin x Giving another approach to Theorem C ...

C 1 e i x c 2 e i x C 1 and c 2 are complex constants c 1 e a ...

Let the roots of the auxiliary equation r2 + a1r + a2 = 0 be a (

Question:

Let the roots of the auxiliary equation r2 + a1r + a2 = 0 be a ( βi. From Problem 25c, it follows, just as in the real case, that y = c1e(a( βi)x + c2e(a-βi)x satisfies (D2 + a1D + a2)y = 0. Show that this solution can be rewritten in the form
Y = C1eax cos βx + C2eax sin βx
Giving another approach to Theorem C.