81y^{''}-18y^'+5y=0,y(1)=1,y^'(1)=7

解

81 y^{^{''}} - 18 y^{^{'}} + 5 y = 0, y (1) = 1, y^{^{'}} (1) = 7

y = e^{\frac{t}{9}} \frac{( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) - 9 e sin ( \frac{2}{9} ) ( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) ) cos ( \frac{2 t}{9} )}{9 e cos ( \frac{2}{9} ) ( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) )} + \frac{( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) sin ( \frac{2 t}{9} )}{- 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} )}

解答ステップ

81 y^{''} (t) - 18 y^{'} (t) + 5 y = 0

線形 ODE を解く:

y = e^{\frac{t}{9}} \frac{( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) - 9 e sin ( \frac{2}{9} ) ( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) ) cos ( \frac{2 t}{9} )}{9 e cos ( \frac{2}{9} ) ( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) )} + \frac{( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) sin ( \frac{2 t}{9} )}{- 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} )}

y = e^{\frac{t}{9}} \frac{( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) - 9 e sin ( \frac{2}{9} ) ( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) ) cos ( \frac{2 t}{9} )}{9 e cos ( \frac{2}{9} ) ( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) )} + \frac{( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) sin ( \frac{2 t}{9} )}{- 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} )}