100y^{''}-140y^'+74y=0,y(3)=6,y^'(3)=5

解答

100 y^{^{''}} - 140 y^{^{'}} + 74 y = 0, y (3) = 6, y^{^{'}} (3) = 5

y = e^{\frac{7 t}{10}} (\frac{( 30 ( cos ^{2} ( \frac{3}{2} ) + sin ^{2} ( \frac{3}{2} ) ) - sin ( \frac{3}{2} ) ( 8 cos ( \frac{3}{2} ) + 30 sin ( \frac{3}{2} ) ) ) cos ( \frac{t}{2} )}{5 e ^{\frac{21}{10}} cos ( \frac{3}{2} ) ( cos ^{2} ( \frac{3}{2} ) + sin ^{2} ( \frac{3}{2} ) )} + \frac{( 8 cos ( \frac{3}{2} ) + 30 sin ( \frac{3}{2} ) ) sin ( \frac{t}{2} )}{5 e ^{\frac{21}{10}} ( cos ^{2} ( \frac{3}{2} ) + sin ^{2} ( \frac{3}{2} ) )})

求解步骤

100 y^{''} (t) - 140 y^{'} (t) + 74 y = 0

求解线性方程:

y = e^{\frac{7 t}{10}} (\frac{( 30 ( cos ^{2} ( \frac{3}{2} ) + sin ^{2} ( \frac{3}{2} ) ) - sin ( \frac{3}{2} ) ( 8 cos ( \frac{3}{2} ) + 30 sin ( \frac{3}{2} ) ) ) cos ( \frac{t}{2} )}{5 e ^{\frac{21}{10}} cos ( \frac{3}{2} ) ( cos ^{2} ( \frac{3}{2} ) + sin ^{2} ( \frac{3}{2} ) )} + \frac{( 8 cos ( \frac{3}{2} ) + 30 sin ( \frac{3}{2} ) ) sin ( \frac{t}{2} )}{5 e ^{\frac{21}{10}} ( cos ^{2} ( \frac{3}{2} ) + sin ^{2} ( \frac{3}{2} ) )})

y = e^{\frac{7 t}{10}} (\frac{( 30 ( cos ^{2} ( \frac{3}{2} ) + sin ^{2} ( \frac{3}{2} ) ) - sin ( \frac{3}{2} ) ( 8 cos ( \frac{3}{2} ) + 30 sin ( \frac{3}{2} ) ) ) cos ( \frac{t}{2} )}{5 e ^{\frac{21}{10}} cos ( \frac{3}{2} ) ( cos ^{2} ( \frac{3}{2} ) + sin ^{2} ( \frac{3}{2} ) )} + \frac{( 8 cos ( \frac{3}{2} ) + 30 sin ( \frac{3}{2} ) ) sin ( \frac{t}{2} )}{5 e ^{\frac{21}{10}} ( cos ^{2} ( \frac{3}{2} ) + sin ^{2} ( \frac{3}{2} ) )})

y^{^{''}} - 26 y^{^{'}} + 233 y = 0

(D^{4} + 2 D^{2}) y = 0

y^{^{''}} + 6 y^{^{'}} + 3 y = 0, y (0) = 0, y^{^{'}} (0) = 1

60 x^{^{''}} + 240 x = 0

y^{^{''}} + (\frac{1}{2}) y = 0