9y^{''}-y=xe^{x/3},y(0)=1,y(0)=0

Lösung

9 y^{^{''}} - y = x e^{\frac{x}{3}}, y (0) = 1, y (0) = 0

y = (- c_{2} + \frac{5}{8}) e^{\frac{x}{3}} + c_{2} e^{- \frac{x}{3}} + \frac{e ^{\frac{x}{3}} x ^{2}}{12} - \frac{1}{8} e^{- \frac{x}{3}} (2 e^{\frac{2 x}{3}} x - 3 e^{\frac{2 x}{3}})

Schritte zur Lösung

9 y^{''} (x) - y = x e^{\frac{x}{3}}

Solve linear ODE:

y = (- c_{2} + \frac{5}{8}) e^{\frac{x}{3}} + c_{2} e^{- \frac{x}{3}} + \frac{e ^{\frac{x}{3}} x ^{2}}{12} - \frac{1}{8} e^{- \frac{x}{3}} (2 e^{\frac{2 x}{3}} x - 3 e^{\frac{2 x}{3}})

y = (- c_{2} + \frac{5}{8}) e^{\frac{x}{3}} + c_{2} e^{- \frac{x}{3}} + \frac{e ^{\frac{x}{3}} x ^{2}}{12} - \frac{1}{8} e^{- \frac{x}{3}} (2 e^{\frac{2 x}{3}} x - 3 e^{\frac{2 x}{3}})

(D^{2} + 1) y = - t

(d^{2} - 4 d + 5) y = (x + 1)^{3}

y^{^{''''}} + 2 y^{^{'''}} + 2 y^{^{''}} + 2 y^{^{'}} + y = x e^{x}

y^{^{''}} - 4 y^{^{'}} + 3 y = (x^{2} - 3 x) e^{x}

y^{^{''}} + 4 y^{^{'}} + 3 y = 50 x \cdot sin (x)