Question

Alguem poderia me ajudar com a E.D.O?$y"-6y'+9y=0 y(0)=0; y'(0)=1$

Answer 1

$y(x) = {e}^{mx} \\ y'(x) = m {e}^{mx} \\ y"(x) = {m}^{2} {e}^{mx}$

${m}^{2} {e}^{mx} - 6m {e}^{mx} + 9 {e}^{mx} = 0$

$y'(0) = m {e}^{0} \\ m = 1$

${e}^{mx} ( {m}^{2} - 6m + 9) = 0 \ \\ {e}^{mx} ≠ 0 \\ {m}^{2} - 6m + 9 = 0 \\ {(m - 3)}^{2} = 0$

$m - 3 = 0 \\ m = 3$

$y(x) = c1 {e}^{mx} + c2.x {e}^{mx} \\ y(x) = c1 {e}^{x} + c2.x {e}^{3x}$