(dr(θ))/(r(θ))=(r(θ)dθ)/(θ^2)

解答

\frac{d r ( θ )}{r ( θ )} = \frac{r ( θ ) d θ}{θ ^{2}}

θ = c_{1} r, θ = \frac{c _{1}}{r}

求解步骤

\frac{d r}{r} = \frac{r d θ}{θ ^{2}}

令

θ

为因变量。除以

d r

\frac{\frac{d θ}{d r}}{r \frac{d θ}{d r}} = \frac{r \frac{d θ}{d r} \cdot \frac{d θ}{d r}}{θ ^{2}}

用

θ^{'} (r)

替代

\frac{d θ}{d r}

\frac{θ ^{^{'}} ( r )}{r θ ^{^{'}} ( r )} = \frac{r θ ^{^{'}} ( r ) θ ^{^{'}} ( r )}{θ ^{2}}

整理

θ^{'} (r) : θ^{'} (r) = \frac{θ}{r}, θ^{'} (r) = - \frac{θ}{r}

θ^{'} (r) = \frac{θ}{r} : θ = c_{1} r

θ^{'} (r) = - \frac{θ}{r} : θ = \frac{c _{1}}{r}

θ = c_{1} r, θ = \frac{c _{1}}{r}

(1 - y^{2}) + x (\frac{d y}{d x}) = 0

y^{^{'}} = 0.2 x y

y^{^{'}} = \frac{y cos ( x )}{1 - 2 y}

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

x^{2} y^{^{'}} = \frac{1}{y} x^{5}, y (0) = 2