y^{''}+10y^'+20y=50sinh(5x)

解答

y^{^{''}} + 10 y^{^{'}} + 20 y = 50 sinh (5 x)

y = c_{1} e^{(- 5 + 5) x} + c_{2} e^{(- 5 - 5) x} + e^{- 5 x + 5 x} \cdot \int 55 e^{5 x - 5 x} sinh (5 x) d x + e^{- 5 x - 5 x} \cdot \int - 55 e^{5 x + 5 x} sinh (5 x) d x

求解步骤

y^{''} (x) + 10 y^{'} (x) + 20 y = 50 sinh (5 x)

求解线性方程:

y = c_{1} e^{(- 5 + 5) x} + c_{2} e^{(- 5 - 5) x} + e^{- 5 x + 5 x} \cdot \int 55 e^{5 x - 5 x} sinh (5 x) d x + e^{- 5 x - 5 x} \cdot \int - 55 e^{5 x + 5 x} sinh (5 x) d x

y = c_{1} e^{(- 5 + 5) x} + c_{2} e^{(- 5 - 5) x} + e^{- 5 x + 5 x} \cdot \int 55 e^{5 x - 5 x} sinh (5 x) d x + e^{- 5 x - 5 x} \cdot \int - 55 e^{5 x + 5 x} sinh (5 x) d x

\frac{d ^{4} y}{d x ^{4}} + 3 \frac{d ^{2} y}{d x ^{2}} - 4 = 0

y^{^{''}} - y^{^{'}} = x + sin (x)

y^{^{''}} + 17 y = 5 e^{x}

y^{^{''}} + 3 y^{^{'}} + 2 y = \frac{1}{( 9 + e ^{x} )}

so l v e f or d, d (x) = lo g_{10} (\frac{x + 4}{x - 1})