Internal problem ID [11234]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first.
Article 28. Summary. Page 59
Problem number: Ex 10.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
\[ \boxed {y-y^{\prime } x -\frac {y {y^{\prime }}^{2}}{x^{2}}=0} \]
✓ Solution by Maple
Time used: 0.344 (sec). Leaf size: 89
dsolve(y(x)=x*diff(y(x),x)+y(x)*diff(y(x),x)^2/x^2,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {i x^{2}}{2} \\ y \left (x \right ) &= \frac {i x^{2}}{2} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\sqrt {c_{1} \left (-4 x^{2}+c_{1} \right )}}{4} \\ y \left (x \right ) &= \frac {\sqrt {c_{1} \left (-4 x^{2}+c_{1} \right )}}{4} \\ y \left (x \right ) &= -\frac {2 \sqrt {c_{1} x^{2}+4}}{c_{1}} \\ y \left (x \right ) &= \frac {2 \sqrt {c_{1} x^{2}+4}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.986 (sec). Leaf size: 244
DSolve[y[x]==x*y'[x]+y[x]*(y'[x])^2/x^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {\sqrt {x^6+4 x^2 y(x)^2} \log \left (\sqrt {x^4+4 y(x)^2}+x^2\right )}{2 x \sqrt {x^4+4 y(x)^2}}+\frac {1}{2} \left (1-\frac {\sqrt {x^6+4 x^2 y(x)^2}}{x \sqrt {x^4+4 y(x)^2}}\right ) \log (y(x))&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {1}{2} \left (\frac {\sqrt {x^6+4 x^2 y(x)^2}}{x \sqrt {x^4+4 y(x)^2}}+1\right ) \log (y(x))-\frac {\sqrt {x^6+4 x^2 y(x)^2} \log \left (\sqrt {x^4+4 y(x)^2}+x^2\right )}{2 x \sqrt {x^4+4 y(x)^2}}&=c_1,y(x)\right ] \\ y(x)\to -\frac {i x^2}{2} \\ y(x)\to \frac {i x^2}{2} \\ \end{align*}