(d^2y)/(dx^2)+3(dy)/(dx)+2y= 1/(1+e^x)

Lösung

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

y = c_{1} e^{- x} + c_{2} e^{- 2 x} + ln (1 + e^{x}) e^{- x} + (- 1 - e^{x} + ln (1 + e^{x})) e^{- 2 x}

Schritte zur Lösung

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

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} e^{- 2 x} + ln (1 + e^{x}) e^{- x} + (- 1 - e^{x} + ln (1 + e^{x})) e^{- 2 x}

y = c_{1} e^{- x} + c_{2} e^{- 2 x} + ln (1 + e^{x}) e^{- x} + (- 1 - e^{x} + ln (1 + e^{x})) e^{- 2 x}

y^{^{''}} - y^{^{'}} + 169 y = 13 sin (13 t)

y^{^{''}} + 4 y = sin (t), y (0) = 0, y^{^{'}} (0) = 0

y^{^{''}} + 5 y^{^{'}} = 4 x + 8

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

y^{^{''}} + 4 y = 4 csc^{2} (2 t)