4y^{''}-y=xe^{x/2},y(0)=1,y^'(0)=0

Lösung

4 y^{^{''}} - y = x e^{\frac{x}{2}}, y (0) = 1, y^{^{'}} (0) = 0

y = (- \frac{1}{4} + \frac{3}{4}) e^{\frac{x}{2}} + \frac{1}{4} e^{- \frac{x}{2}} + \frac{e ^{\frac{x}{2}} x ^{2}}{8} + \frac{- e ^{\frac{x}{2}} x + e ^{\frac{x}{2}}}{4}

Schritte zur Lösung

4 y^{''} (x) - y = x e^{\frac{x}{2}}

Solve linear ODE:

y = (- \frac{1}{4} + \frac{3}{4}) e^{\frac{x}{2}} + \frac{1}{4} e^{- \frac{x}{2}} + \frac{e ^{\frac{x}{2}} x ^{2}}{8} + \frac{- e ^{\frac{x}{2}} x + e ^{\frac{x}{2}}}{4}

y = (- \frac{1}{4} + \frac{3}{4}) e^{\frac{x}{2}} + \frac{1}{4} e^{- \frac{x}{2}} + \frac{e ^{\frac{x}{2}} x ^{2}}{8} + \frac{- e ^{\frac{x}{2}} x + e ^{\frac{x}{2}}}{4}

y^{^{''}} + 9 y = g (t), y (0) = 1, y^{^{'}} (0) = 0

y^{^{''}} - 4 y^{^{'}} - 12 y = 3 x^{2} - 5 x + 4 - 3 e^{- 3 x}

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

y^{^{''}} - y^{^{'}} - 6 y = 15 e^{3 t}, y (0) = 2, y^{^{'}} (0) = - 6

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