100y^{''}+60y^'+90y=0,y(1)=9,y^'(1)=2

解答

100 y^{^{''}} + 60 y^{^{'}} + 90 y = 0, y (1) = 9, y^{^{'}} (1) = 2

y = e^{- \frac{3 t}{10}} \frac{( 27 e ^{\frac{3}{10}} ( - sin ( \frac{9}{10} ) + 3 cos ( \frac{9}{10} ) + 3 sin ^{2} ( \frac{9}{10} ) sec ( \frac{9}{10} ) + cos ( \frac{9}{10} ) sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) ) - sin ( \frac{9}{10} ) ( 20 e ^{\frac{3}{10}} + 27 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) cos ( \frac{9}{10} ) + 81 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) ) ) cos ( \frac{9 t}{10} )}{3 cos ( \frac{9}{10} ) ( - sin ( \frac{9}{10} ) + 3 cos ( \frac{9}{10} ) + 3 sin ^{2} ( \frac{9}{10} ) sec ( \frac{9}{10} ) + cos ( \frac{9}{10} ) sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) )} + \frac{( 20 e ^{\frac{3}{10}} + 27 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) cos ( \frac{9}{10} ) + 81 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) ) sin ( \frac{9 t}{10} )}{3 ( - sin ( \frac{9}{10} ) + 3 cos ( \frac{9}{10} ) + 3 sin ^{2} ( \frac{9}{10} ) sec ( \frac{9}{10} ) + cos ( \frac{9}{10} ) sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) )}

求解步骤

100 y^{''} (t) + 60 y^{'} (t) + 90 y = 0

求解线性方程:

y = e^{- \frac{3 t}{10}} \frac{( 27 e ^{\frac{3}{10}} ( - sin ( \frac{9}{10} ) + 3 cos ( \frac{9}{10} ) + 3 sin ^{2} ( \frac{9}{10} ) sec ( \frac{9}{10} ) + cos ( \frac{9}{10} ) sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) ) - sin ( \frac{9}{10} ) ( 20 e ^{\frac{3}{10}} + 27 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) cos ( \frac{9}{10} ) + 81 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) ) ) cos ( \frac{9 t}{10} )}{3 cos ( \frac{9}{10} ) ( - sin ( \frac{9}{10} ) + 3 cos ( \frac{9}{10} ) + 3 sin ^{2} ( \frac{9}{10} ) sec ( \frac{9}{10} ) + cos ( \frac{9}{10} ) sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) )} + \frac{( 20 e ^{\frac{3}{10}} + 27 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) cos ( \frac{9}{10} ) + 81 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) ) sin ( \frac{9 t}{10} )}{3 ( - sin ( \frac{9}{10} ) + 3 cos ( \frac{9}{10} ) + 3 sin ^{2} ( \frac{9}{10} ) sec ( \frac{9}{10} ) + cos ( \frac{9}{10} ) sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) )}

y = e^{- \frac{3 t}{10}} \frac{( 27 e ^{\frac{3}{10}} ( - sin ( \frac{9}{10} ) + 3 cos ( \frac{9}{10} ) + 3 sin ^{2} ( \frac{9}{10} ) sec ( \frac{9}{10} ) + cos ( \frac{9}{10} ) sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) ) - sin ( \frac{9}{10} ) ( 20 e ^{\frac{3}{10}} + 27 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) cos ( \frac{9}{10} ) + 81 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) ) ) cos ( \frac{9 t}{10} )}{3 cos ( \frac{9}{10} ) ( - sin ( \frac{9}{10} ) + 3 cos ( \frac{9}{10} ) + 3 sin ^{2} ( \frac{9}{10} ) sec ( \frac{9}{10} ) + cos ( \frac{9}{10} ) sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) )} + \frac{( 20 e ^{\frac{3}{10}} + 27 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) cos ( \frac{9}{10} ) + 81 e ^{\frac{3}{10}} sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) ) sin ( \frac{9 t}{10} )}{3 ( - sin ( \frac{9}{10} ) + 3 cos ( \frac{9}{10} ) + 3 sin ^{2} ( \frac{9}{10} ) sec ( \frac{9}{10} ) + cos ( \frac{9}{10} ) sec ( \frac{9}{10} ) sin ( \frac{9}{10} ) )}

x^{^{'''}} + 9 \cdot x^{^{''}} + x^{^{'}} + 9 x = 0

(D^{3} - 3 D^{2} - 10 D) y = 0

y^{^{''}} + 3 y^{^{'}} - y = 0, y (1) = 2, y^{^{'}} (1) = 0

y^{^{''}} - 9 y = 0, y (0) = 5, y^{^{'}} (0) = 9

\frac{d ^{2} y}{d t ^{2}} + 6 \frac{d y}{d t} + 9 y = 0, y (0) = 2