Internal problem ID [9006]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 671.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
\[ \boxed {y^{\prime }-\frac {\left (y^{2} x +1\right )^{2}}{y x^{4}}=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 197
dsolve(diff(y(x),x) = (x*y(x)^2+1)^2/y(x)/x^4,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, x +1}{x^{2}}} \sqrt {-\left ({\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \right ) x \left (\left (2-\sqrt {2}\, x \right ) {\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \left (\sqrt {2}\, x +2\right )\right ) {\mathrm e}^{-\frac {2}{x^{2}}} {\mathrm e}^{-\frac {2 \sqrt {2}}{x}}}}{2 x \left ({\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \right )} \\ y \left (x \right ) &= \frac {\sqrt {2}\, {\mathrm e}^{\frac {\sqrt {2}\, x +1}{x^{2}}} \sqrt {-\left ({\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \right ) x \left (\left (2-\sqrt {2}\, x \right ) {\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \left (\sqrt {2}\, x +2\right )\right ) {\mathrm e}^{-\frac {2}{x^{2}}} {\mathrm e}^{-\frac {2 \sqrt {2}}{x}}}}{2 x \left ({\mathrm e}^{\frac {2 \sqrt {2}}{x}}+c_{1} \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 14.007 (sec). Leaf size: 206
DSolve[y'[x] == (1 + x*y[x]^2)^2/(x^4*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\frac {-\sqrt {2} x+\left (\sqrt {2} x-2\right ) e^{\frac {2 \sqrt {2} (1+c_1 x)}{x}}-2}{x}}}{\sqrt {2} \sqrt {1+e^{\frac {2 \sqrt {2} (1+c_1 x)}{x}}}} \\ y(x)\to \frac {\sqrt {\frac {-\sqrt {2} x+\left (\sqrt {2} x-2\right ) e^{\frac {2 \sqrt {2} (1+c_1 x)}{x}}-2}{x}}}{\sqrt {2} \sqrt {1+e^{\frac {2 \sqrt {2} (1+c_1 x)}{x}}}} \\ y(x)\to -\sqrt {-\frac {1}{x}-\frac {1}{\sqrt {2}}} \\ y(x)\to \sqrt {-\frac {1}{x}-\frac {1}{\sqrt {2}}} \\ \end{align*}