Internal problem ID [7993]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 413.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, class G], _rational]
Solve \begin {gather*} \boxed {x \left (y^{\prime }\right )^{2}+y y^{\prime }-x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 337
dsolve(x*diff(y(x),x)^2+y(x)*diff(y(x),x)-x^2 = 0,y(x), singsol=all)
\begin{align*} \int _{\textit {\_b}}^{x}-\frac {-y \relax (x )+\sqrt {4 \textit {\_a}^{3}+y \relax (x )^{2}}}{\left (-4 y \relax (x )+\sqrt {4 \textit {\_a}^{3}+y \relax (x )^{2}}\right ) \textit {\_a}}d \textit {\_a} +\int _{}^{y \relax (x )}\frac {12 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (-4 \textit {\_f} +\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}\right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \sqrt {4 x^{3}+\textit {\_f}^{2}}-48 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (-4 \textit {\_f} +\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}\right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \textit {\_f} +2}{-4 \textit {\_f} +\sqrt {4 x^{3}+\textit {\_f}^{2}}}d \textit {\_f} +c_{1} = 0 \\ \int _{\textit {\_b}}^{x}-\frac {y \relax (x )+\sqrt {4 \textit {\_a}^{3}+y \relax (x )^{2}}}{\left (\sqrt {4 \textit {\_a}^{3}+y \relax (x )^{2}}+4 y \relax (x )\right ) \textit {\_a}}d \textit {\_a} +\int _{}^{y \relax (x )}-\frac {2 \left (6 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}+4 \textit {\_f} \right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \sqrt {4 x^{3}+\textit {\_f}^{2}}+24 \left (\int _{\textit {\_b}}^{x}\frac {\textit {\_a}^{2}}{\left (\sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}+4 \textit {\_f} \right )^{2} \sqrt {4 \textit {\_a}^{3}+\textit {\_f}^{2}}}d \textit {\_a} \right ) \textit {\_f} +1\right )}{\sqrt {4 x^{3}+\textit {\_f}^{2}}+4 \textit {\_f}}d \textit {\_f} +c_{1} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.275 (sec). Leaf size: 673
DSolve[-x^2 + y[x]*y'[x] + x*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\int \left (\frac {4 \sqrt {4 x^3+y(x)^2} x^2}{5 y(x) \left (4 x^3-15 y(x)^2\right )}+\frac {16 x^2}{5 \left (4 x^3-15 y(x)^2\right )}-\frac {\sqrt {4 x^3+y(x)^2}}{5 y(x) x}+\frac {1}{5 x}\right )dx+\int \left (\frac {8 y(x)}{15 y(x)^2-4 x^3}-\int \left (-\frac {4 \sqrt {4 x^3+y(x)^2} x^2}{5 y(x)^2 \left (4 x^3-15 y(x)^2\right )}+\frac {24 \sqrt {4 x^3+y(x)^2} x^2}{\left (4 x^3-15 y(x)^2\right )^2}+\frac {4 x^2}{5 \left (4 x^3-15 y(x)^2\right ) \sqrt {4 x^3+y(x)^2}}+\frac {96 y(x) x^2}{\left (4 x^3-15 y(x)^2\right )^2}+\frac {\sqrt {4 x^3+y(x)^2}}{5 y(x)^2 x}-\frac {1}{5 \sqrt {4 x^3+y(x)^2} x}\right )dx+\frac {2 \sqrt {4 x^3+y(x)^2}}{15 y(x)^2-4 x^3}\right )dy(x)=c_1,y(x)\right ] \\ \text {Solve}\left [\int \left (-\frac {4 \sqrt {4 x^3+y(x)^2} x^2}{5 y(x) \left (4 x^3-15 y(x)^2\right )}+\frac {16 x^2}{5 \left (4 x^3-15 y(x)^2\right )}+\frac {\sqrt {4 x^3+y(x)^2}}{5 y(x) x}+\frac {1}{5 x}\right )dx+\int \left (\frac {8 y(x)}{15 y(x)^2-4 x^3}-\int \left (\frac {4 \sqrt {4 x^3+y(x)^2} x^2}{5 y(x)^2 \left (4 x^3-15 y(x)^2\right )}-\frac {24 \sqrt {4 x^3+y(x)^2} x^2}{\left (4 x^3-15 y(x)^2\right )^2}-\frac {4 x^2}{5 \left (4 x^3-15 y(x)^2\right ) \sqrt {4 x^3+y(x)^2}}+\frac {96 y(x) x^2}{\left (4 x^3-15 y(x)^2\right )^2}-\frac {\sqrt {4 x^3+y(x)^2}}{5 y(x)^2 x}+\frac {1}{5 \sqrt {4 x^3+y(x)^2} x}\right )dx-\frac {2 \sqrt {4 x^3+y(x)^2}}{15 y(x)^2-4 x^3}\right )dy(x)=c_1,y(x)\right ] \\ \end{align*}