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y^{1/2}(dy)/(dx)+y^{3/2}=1,y(0)=9

解答

y^{\frac{1}{2}} \frac{d y}{d x} + y^{\frac{3}{2}} = 1, y (0) = 9

解答

y = 3 676 e^{- 3 x} + 52 e^{- \frac{3 x}{2}} + 1

求解步骤

y^{\frac{1}{2}} \frac{d y}{d x} + y^{\frac{3}{2}} = 1

替代

\frac{d y}{d x}

与

y^{'} (x)

y^{\frac{1}{2}} y^{^{'}} (x) + y^{\frac{3}{2}} = 1

求解

x, y^{'} (x) = \frac{d y}{d x} = \frac{1}{\frac{d x}{d y}} = \frac{1}{x ^{^{'}}}

y^{\frac{1}{2}} (\frac{1}{x ^{^{^{'}}}}) + y^{\frac{3}{2}} = 1

整理

x^{^{'}} : x^{^{'}} = - \frac{y ^{\frac{1}{2}}}{y ^{\frac{3}{2}} - 1}

若

f^{^{'}} (x) = g (x)

则

f (x) = \int g (x) d x

x = \int - \frac{y ^{\frac{1}{2}}}{y ^{\frac{3}{2}} - 1} d y

\int - \frac{y ^{\frac{1}{2}}}{y ^{\frac{3}{2}} - 1} d y = - \frac{2}{3} ln (y^{\frac{3}{2}} - 1) + c_{1}

x = - \frac{2}{3} ln (y^{\frac{3}{2}} - 1) + c_{1}

应用初始条件:

x = - \frac{2}{3} ln (y^{\frac{3}{2}} - 1) + \frac{2 ln ( 26 )}{3}

整理

y : y = 3 676 e^{- 3 x} + 52 e^{- \frac{3 x}{2}} + 1

使用初始条件来验证解

y = 3 676 e^{- 3 x} + 52 e^{- \frac{3 x}{2}} + 1

作图

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