y^{''}+10/3 y^'+ay=0,y(0)=1,y^'(0)= 5/3

解答

y^{^{''}} + \frac{10}{3} y^{^{'}} + a y = 0, y (0) = 1, y^{^{'}} (0) = \frac{5}{3}

y = e^{\frac{t ( - 5 + - 9 a + 25 )}{3}} (1 + \frac{- - 9 a + 25 + 10}{2 25 - 9 a}) - \frac{e ^{- \frac{t ( - 9 a + 25 + 5 )}{3}} ( - - 9 a + 25 + 10 )}{2 25 - 9 a}

求解步骤

y^{''} (t) + \frac{10}{3} y^{'} (t) + a y = 0

求解线性方程:

y = e^{\frac{t ( - 5 + - 9 a + 25 )}{3}} (1 + \frac{- - 9 a + 25 + 10}{2 25 - 9 a}) - \frac{e ^{- \frac{t ( - 9 a + 25 + 5 )}{3}} ( - - 9 a + 25 + 10 )}{2 25 - 9 a}

y = e^{\frac{t ( - 5 + - 9 a + 25 )}{3}} (1 + \frac{- - 9 a + 25 + 10}{2 25 - 9 a}) - \frac{e ^{- \frac{t ( - 9 a + 25 + 5 )}{3}} ( - - 9 a + 25 + 10 )}{2 25 - 9 a}

u^{^{''}} + u^{^{'}} = 0

9 y^{^{''}} - 30 y^{^{'}} - 24 y = 0, y (0) = 0, y^{^{'}} (0) = 1

\frac{d ^{2} y}{d x ^{2}} - 7 \frac{d y}{d x} + 6 y = 0

2 y^{^{''}} + 5 y^{^{'}} - 3 y = 0, y (0) = 5, y^{^{''}} (0) = 13

x^{^{''}} + c x = 0