y^{'''}+4y^'=tan(x)

解答

y^{^{'''}} + 4 y^{^{'}} = tan (x)

y = c_{1} + c_{2} cos (2 x) + c_{3} sin (2 x) - \frac{1}{4} ln (cos (x)) - \frac{cos ( 2 x ) ( tan ( x ) sin ( 2 x ) + ln ( cos ^{2} ( x ) ) )}{8} - \frac{1}{4} sin (2 x) (x - \frac{1}{2} sin (2 x))

求解步骤

y^{'''} (x) + 4 y^{'} (x) = tan (x)

求解线性方程:

y = c_{1} + c_{2} cos (2 x) + c_{3} sin (2 x) - \frac{1}{4} ln (cos (x)) - \frac{cos ( 2 x ) ( tan ( x ) sin ( 2 x ) + ln ( cos ^{2} ( x ) ) )}{8} - \frac{1}{4} sin (2 x) (x - \frac{1}{2} sin (2 x))

y = c_{1} + c_{2} cos (2 x) + c_{3} sin (2 x) - \frac{1}{4} ln (cos (x)) - \frac{cos ( 2 x ) ( tan ( x ) sin ( 2 x ) + ln ( cos ^{2} ( x ) ) )}{8} - \frac{1}{4} sin (2 x) (x - \frac{1}{2} sin (2 x))

y^{^{''}} - 10 y^{^{'}} + 25 y = 25 x + 10

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

\frac{d ^{2} y}{d x ^{2}} - \frac{d y}{d x} = sin (x) - cos (x)

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

x^{^{''}} + 5 x^{^{'}} + 17 x = 12 cos (3 t)