y^{''}-2y^'+2y=3x+e^xtan(x)

解

y^{^{''}} - 2 y^{^{'}} + 2 y = 3 x + e^{x} tan (x)

y = e^{x} (c_{1} cos (x) + c_{2} sin (x)) + e^{x} cos (x) (\frac{3}{2} e^{- x} x cos (x) + \frac{3}{2} e^{- x} x sin (x) + \frac{3}{2} e^{- x} cos (x) - ln (tan (x) + sec (x)) + sin (x)) + e^{x} sin (x) \cdot \int \frac{cos ( x ) ( 3 x + e ^{x} tan ( x ) )}{e ^{x}} d x

解答ステップ

y^{''} (x) - 2 y^{'} (x) + 2 y = 3 x + e^{x} tan (x)

線形 ODE を解く:

y = e^{x} (c_{1} cos (x) + c_{2} sin (x)) + e^{x} cos (x) (\frac{3}{2} e^{- x} x cos (x) + \frac{3}{2} e^{- x} x sin (x) + \frac{3}{2} e^{- x} cos (x) - ln (tan (x) + sec (x)) + sin (x)) + e^{x} sin (x) \cdot \int \frac{cos ( x ) ( 3 x + e ^{x} tan ( x ) )}{e ^{x}} d x

y = e^{x} (c_{1} cos (x) + c_{2} sin (x)) + e^{x} cos (x) (\frac{3}{2} e^{- x} x cos (x) + \frac{3}{2} e^{- x} x sin (x) + \frac{3}{2} e^{- x} cos (x) - ln (tan (x) + sec (x)) + sin (x)) + e^{x} sin (x) \cdot \int \frac{cos ( x ) ( 3 x + e ^{x} tan ( x ) )}{e ^{x}} d x