1/2 y^{''}+2y=3tan(2t)-1/3 e^{4t}

解

\frac{1}{2} y^{^{''}} + 2 y = 3 tan (2 t) - \frac{1}{3} e^{4 t}

y = c_{1} cos (2 t) + c_{2} sin (2 t) + \frac{- 45 cos ( 2 t ) ln ( tan ( 2 t ) + sec ( 2 t ) ) - e ^{4 t} cos ^{2} ( 2 t ) - e ^{4 t} sin ^{2} ( 2 t )}{30}

解答ステップ

\frac{1}{2} y^{''} (t) + 2 y = 3 tan (2 t) - \frac{1}{3} e^{4 t}

線形 ODE を解く:

y = c_{1} cos (2 t) + c_{2} sin (2 t) + \frac{- 45 cos ( 2 t ) ln ( tan ( 2 t ) + sec ( 2 t ) ) - e ^{4 t} cos ^{2} ( 2 t ) - e ^{4 t} sin ^{2} ( 2 t )}{30}

y = c_{1} cos (2 t) + c_{2} sin (2 t) + \frac{- 45 cos ( 2 t ) ln ( tan ( 2 t ) + sec ( 2 t ) ) - e ^{4 t} cos ^{2} ( 2 t ) - e ^{4 t} sin ^{2} ( 2 t )}{30}