y^{''}+9y^'=t^2e^{3t}+6

Lösung

y^{^{''}} + 9 y^{^{'}} = t^{2} e^{3 t} + 6

y = c_{1} + c_{2} e^{- 9 t} + \frac{e ^{3 t} t ^{2}}{36} - \frac{5 e ^{3 t} t}{216} + \frac{7}{864} e^{3 t} + \frac{2 t}{3}

Schritte zur Lösung

y^{''} (t) + 9 y^{'} (t) = t^{2} e^{3 t} + 6

Solve linear ODE:

y = c_{1} + c_{2} e^{- 9 t} + \frac{e ^{3 t} t ^{2}}{36} - \frac{5 e ^{3 t} t}{216} + \frac{7}{864} e^{3 t} + \frac{2 t}{3}

y = c_{1} + c_{2} e^{- 9 t} + \frac{e ^{3 t} t ^{2}}{36} - \frac{5 e ^{3 t} t}{216} + \frac{7}{864} e^{3 t} + \frac{2 t}{3}

\frac{d ^{2} y}{d t ^{2}} - 12 \frac{d y}{d t} + 36 y = 7 + 4 t

l in e a r y^{^{''}} + y^{^{'}} + 2 y = sin (x)

y^{^{''}} - 2 y^{^{'}} + 2 y = e^{(2 x)} \cdot sin (x)

y^{^{''}} - 15 y^{^{'}} + 54 y = e^{a x}

y^{^{'''}} + 9 y^{^{''}} + y^{^{'}} + 9 y = 16 e^{x} - 9