81y^{''}+180y^'+125y=0,y(3)=6,y^'(3)=6

解

81 y^{^{''}} + 180 y^{^{'}} + 125 y = 0, y (3) = 6, y^{^{'}} (3) = 6

y = e^{- \frac{10 t}{9}} - \frac{( - 30 e ^{\frac{10}{3}} ( sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + cos ( \frac{5}{3} ) ) - sin ( \frac{5}{3} ) ( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) ) cos ( \frac{5 t}{9} )}{5 cos ( \frac{5}{3} ) ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )} - \frac{( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) sin ( \frac{5 t}{9} )}{5 ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )}

解答ステップ

81 y^{''} (t) + 180 y^{'} (t) + 125 y = 0

線形 ODE を解く:

y = e^{- \frac{10 t}{9}} - \frac{( - 30 e ^{\frac{10}{3}} ( sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + cos ( \frac{5}{3} ) ) - sin ( \frac{5}{3} ) ( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) ) cos ( \frac{5 t}{9} )}{5 cos ( \frac{5}{3} ) ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )} - \frac{( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) sin ( \frac{5 t}{9} )}{5 ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )}

y = e^{- \frac{10 t}{9}} - \frac{( - 30 e ^{\frac{10}{3}} ( sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + cos ( \frac{5}{3} ) ) - sin ( \frac{5}{3} ) ( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) ) cos ( \frac{5 t}{9} )}{5 cos ( \frac{5}{3} ) ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )} - \frac{( - 54 e ^{\frac{10}{3}} - 60 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) cos ( \frac{5}{3} ) - 30 e ^{\frac{10}{3}} sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) ) sin ( \frac{5 t}{9} )}{5 ( cos ( \frac{5}{3} ) + sin ^{2} ( \frac{5}{3} ) sec ( \frac{5}{3} ) - 2 sin ( \frac{5}{3} ) + 2 cos ( \frac{5}{3} ) sec ( \frac{5}{3} ) sin ( \frac{5}{3} ) )}