y^{''}-y^'-6y=12x-e^x,y(0)=1,y^'(0)=-2

Lösung

y^{^{''}} - y^{^{'}} - 6 y = 12 x - e^{x}, y (0) = 1, y^{^{'}} (0) = - 2

y = (- \frac{1}{3} + \frac{1}{2}) e^{3 x} + \frac{1}{3} e^{- 2 x} - 2 x + \frac{1}{3} + \frac{1}{6} e^{x}

Schritte zur Lösung

y^{''} (x) - y^{'} (x) - 6 y = 12 x - e^{x}

Solve linear ODE:

y = (- \frac{1}{3} + \frac{1}{2}) e^{3 x} + \frac{1}{3} e^{- 2 x} - 2 x + \frac{1}{3} + \frac{1}{6} e^{x}

y = (- \frac{1}{3} + \frac{1}{2}) e^{3 x} + \frac{1}{3} e^{- 2 x} - 2 x + \frac{1}{3} + \frac{1}{6} e^{x}

y^{^{''}} - 3 y^{^{'}} + 2 y = \frac{e ^{2 x}}{e ^{2 x} + 1}

so l v e f or D, D (t) = 5 cos (\frac{π}{6} t + \frac{7 π}{6}) + 3

y^{^{''}} - 10 y^{^{'}} + 25 y = e^{5 t} ln (t)

(D^{2} + 6 D + 9) y = \frac{16 e ^{- 3 x}}{x ^{2} + 1}

y^{^{''}} + 2 y^{^{'}} + 2 y = 5 sin (t) + 5 cos (t)