(d^3+3d^2-4)y=sinh^2(x)+7

解答

(d^{3} + 3 d^{2} - 4) y = sinh^{2} (x) + 7

y = c_{1} e^{x} + c_{2} e^{- 2 x} + c_{3} x e^{- 2 x} - \frac{1}{108} e^{- 2 x} + \frac{1}{36} e^{2 x} - \frac{13}{18} + e^{- 2 x} \cdot \int \frac{e ^{2 x} ( 3 x - 1 ) ( sinh ^{2} ( x ) + 7 )}{9} d x + e^{- 2 x} x \cdot \int - \frac{e ^{2 x} ( sinh ^{2} ( x ) + 7 )}{3} d x

求解步骤

(d^{3} + 3 d^{2} - 4) y = sinh^{2} (x) + 7

求解线性方程:

y = c_{1} e^{x} + c_{2} e^{- 2 x} + c_{3} x e^{- 2 x} - \frac{1}{108} e^{- 2 x} + \frac{1}{36} e^{2 x} - \frac{13}{18} + e^{- 2 x} \cdot \int \frac{e ^{2 x} ( 3 x - 1 ) ( sinh ^{2} ( x ) + 7 )}{9} d x + e^{- 2 x} x \cdot \int - \frac{e ^{2 x} ( sinh ^{2} ( x ) + 7 )}{3} d x

y = c_{1} e^{x} + c_{2} e^{- 2 x} + c_{3} x e^{- 2 x} - \frac{1}{108} e^{- 2 x} + \frac{1}{36} e^{2 x} - \frac{13}{18} + e^{- 2 x} \cdot \int \frac{e ^{2 x} ( 3 x - 1 ) ( sinh ^{2} ( x ) + 7 )}{9} d x + e^{- 2 x} x \cdot \int - \frac{e ^{2 x} ( sinh ^{2} ( x ) + 7 )}{3} d x

(d^{2} + 6 d + 9) y = 5 e^{3} x

(d^{3} - 2 d^{2} - 3 d + 10) y = 40 cos (x)

(d^{2} + 4 d - 12) \cdot y = (x - 1) \cdot e^{2 x}

(d^{2} + 2 d + 10) y = 25 x^{2} + 3

(d^{2} + 4 d) y = 12