Internal problem ID [8251]
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]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (x y^{2}+1\right )^{2}}{y x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.076 (sec). Leaf size: 237
dsolve(diff(y(x),x) = (x*y(x)^2+1)^2/y(x)/x^4,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {\sqrt {2}\, \sqrt {-\left (c_{1} {\mathrm e}^{\frac {-1-x \sqrt {2}}{x^{2}}}+{\mathrm e}^{\frac {-1+x \sqrt {2}}{x^{2}}}\right ) x \left (c_{1} \left (x \sqrt {2}+2\right ) {\mathrm e}^{\frac {-1-x \sqrt {2}}{x^{2}}}+\left (2-x \sqrt {2}\right ) {\mathrm e}^{\frac {-1+x \sqrt {2}}{x^{2}}}\right )}}{2 x \left (c_{1} {\mathrm e}^{\frac {-1-x \sqrt {2}}{x^{2}}}+{\mathrm e}^{\frac {-1+x \sqrt {2}}{x^{2}}}\right )} \\ y \relax (x ) = \frac {\sqrt {2}\, \sqrt {-\left (c_{1} {\mathrm e}^{\frac {-1-x \sqrt {2}}{x^{2}}}+{\mathrm e}^{\frac {-1+x \sqrt {2}}{x^{2}}}\right ) x \left (c_{1} \left (x \sqrt {2}+2\right ) {\mathrm e}^{\frac {-1-x \sqrt {2}}{x^{2}}}+\left (2-x \sqrt {2}\right ) {\mathrm e}^{\frac {-1+x \sqrt {2}}{x^{2}}}\right )}}{2 x \left (c_{1} {\mathrm e}^{\frac {-1-x \sqrt {2}}{x^{2}}}+{\mathrm e}^{\frac {-1+x \sqrt {2}}{x^{2}}}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 7.722 (sec). Leaf size: 162
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}}}} \\ \end{align*}