y^{''}+2y^'+y=(e^{-x})/(1+x^2)

Lösung

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

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

Schritte zur Lösung

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

Solve linear ODE:

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

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

y^{^{''}} - 8 y^{^{'}} + 7 = 0

y^{^{''}} + 8 y^{^{'}} + 7 y = e^{- 2 x}

y^{^{''}} - y^{^{'}} = 2 x - 3

y^{^{''}} - y^{^{'}} = 2 x + 4

y^{^{''}} + 2 y^{^{'}} + 5 = 0