Internal problem ID [8073]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 493.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {\left (y^{2}-2 a x +a^{2}\right ) \left (y^{\prime }\right )^{2}+2 a y y^{\prime }+y^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.391 (sec). Leaf size: 128
dsolve((y(x)^2-2*a*x+a^2)*diff(y(x),x)^2+2*a*y(x)*diff(y(x),x)+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ \left [x \left (\textit {\_T} \right ) = \frac {\arctanh \left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )^{2} \sqrt {\textit {\_T}^{2}+1}\, a^{2}-2 \arctanh \left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right ) \sqrt {\textit {\_T}^{2}+1}\, c_{1} a -2 \arctanh \left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right ) a^{2}+c_{1}^{2} \sqrt {\textit {\_T}^{2}+1}+a^{2} \sqrt {\textit {\_T}^{2}+1}+2 a c_{1}}{2 a \sqrt {\textit {\_T}^{2}+1}}, y \left (\textit {\_T} \right ) = -\frac {\left (a \arctanh \left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )-c_{1}\right ) \textit {\_T}}{\sqrt {\textit {\_T}^{2}+1}}\right ] \\ \end{align*}
✓ Solution by Mathematica
Time used: 31.993 (sec). Leaf size: 97
DSolve[y[x]^2 + 2*a*y[x]*y'[x] + (a^2 - 2*a*x + y[x]^2)*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=\frac {a^2 K[1]^2+2 a K[1] y(K[1])+K[1]^2 y(K[1])^2+y(K[1])^2}{2 a K[1]^2},y(x)=-\frac {a K[1] \tanh ^{-1}\left (\sqrt {K[1]^2+1}\right )}{\sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}}\right \},\{y(x),K[1]\}\right ] \]