1/2 y^{''}+2y=2tan(2t)-1/2 e^{2t}

解

\frac{1}{2} y^{^{''}} + 2 y = 2 tan (2 t) - \frac{1}{2} e^{2 t}

y = c_{1} cos (2 t) + c_{2} sin (2 t) + \frac{- 8 cos ( 2 t ) ln ( tan ( 2 t ) + sec ( 2 t ) ) - e ^{2 t} cos ^{2} ( 2 t ) - e ^{2 t} sin ^{2} ( 2 t )}{8}

解答ステップ

\frac{1}{2} y^{''} (t) + 2 y = 2 tan (2 t) - \frac{1}{2} e^{2 t}

線形 ODE を解く:

y = c_{1} cos (2 t) + c_{2} sin (2 t) + \frac{- 8 cos ( 2 t ) ln ( tan ( 2 t ) + sec ( 2 t ) ) - e ^{2 t} cos ^{2} ( 2 t ) - e ^{2 t} sin ^{2} ( 2 t )}{8}

y = c_{1} cos (2 t) + c_{2} sin (2 t) + \frac{- 8 cos ( 2 t ) ln ( tan ( 2 t ) + sec ( 2 t ) ) - e ^{2 t} cos ^{2} ( 2 t ) - e ^{2 t} sin ^{2} ( 2 t )}{8}