4y^{''}+8y^'+5y=e^{-x}sec(x/2)

Lösung

4 y^{^{''}} + 8 y^{^{'}} + 5 y = e^{- x} sec (\frac{x}{2})

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

Schritte zur Lösung

4 y^{''} (x) + 8 y^{'} (x) + 5 y = e^{- x} sec (\frac{x}{2})

Solve linear ODE:

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

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

y^{^{''}} + 2 y^{^{'}} = 24 x + e^{- 2 x}

x^{^{''}} + 16 x = sin (t)

\frac{d ^{2} y}{d t ^{2}} + 8 \frac{d y}{d t} - 20 y = 5 e^{- 3 t}

y^{^{''}} + y^{^{'}} + 4 = - 20 x

y^{^{''}} - 6 y^{^{'}} + 9 y = 15 x + 3