Internal problem ID [8071]
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')`]
\[ \boxed {\left (y^{2}-2 a x +a^{2}\right ) {y^{\prime }}^{2}+2 a y y^{\prime }+y^{2}=0} \]
✓ Solution by Maple
Time used: 0.328 (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 \left (x \right ) = 0 \\ \left [x \left (\textit {\_T} \right ) = \frac {{\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )}^{2} \sqrt {\textit {\_T}^{2}+1}\, a^{2}-2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right ) \sqrt {\textit {\_T}^{2}+1}\, c_{1} a -2 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right ) a^{2}+\sqrt {\textit {\_T}^{2}+1}\, c_{1}^{2}+a^{2} \sqrt {\textit {\_T}^{2}+1}+2 c_{1} a}{2 a \sqrt {\textit {\_T}^{2}+1}}, y \left (\textit {\_T} \right ) = -\frac {\left (a \,\operatorname {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: 52.585 (sec). Leaf size: 514
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]
\begin{align*} \text {Solve}\left [\left \{y(x)=\frac {-\sqrt {-a K[1]^2 \left (a K[1]^2-2 x K[1]^2-2 x\right )}-a K[1]}{K[1]^2+1},x=\frac {a K[1]^2 \text {arctanh}\left (\sqrt {K[1]^2+1}\right )^2+a \text {arctanh}\left (\sqrt {K[1]^2+1}\right )^2-2 a \sqrt {K[1]^2+1} \text {arctanh}\left (\sqrt {K[1]^2+1}\right )+2 a c_1 K[1]^2 \text {arctanh}\left (\sqrt {K[1]^2+1}\right )+2 a c_1 \text {arctanh}\left (\sqrt {K[1]^2+1}\right )+a K[1]^2+a c_1{}^2 K[1]^2-2 a c_1 \sqrt {K[1]^2+1}+a+a c_1{}^2}{2 \left (K[1]^2+1\right )}\right \},\{y(x),K[1]\}\right ] \\ \text {Solve}\left [\left \{y(x)=\frac {\sqrt {-a K[2]^2 \left (a K[2]^2-2 x K[2]^2-2 x\right )}-a K[2]}{K[2]^2+1},x=\frac {a K[2]^2+a K[2]^2 \log ^2(K[2])+a \log ^2(K[2])+a K[2]^2 \log ^2\left (\sqrt {K[2]^2+1}+1\right )+a \log ^2\left (\sqrt {K[2]^2+1}+1\right )+2 a \sqrt {K[2]^2+1} \log (K[2])-2 a K[2]^2 \log (K[2]) \log \left (\sqrt {K[2]^2+1}+1\right )-2 a \log (K[2]) \log \left (\sqrt {K[2]^2+1}+1\right )-2 a \sqrt {K[2]^2+1} \log \left (\sqrt {K[2]^2+1}+1\right )+a c_1{}^2 K[2]^2-2 a c_1 \sqrt {K[2]^2+1}-2 a c_1 K[2]^2 \log (K[2])-2 a c_1 \log (K[2])+2 a c_1 K[2]^2 \log \left (\sqrt {K[2]^2+1}+1\right )+2 a c_1 \log \left (\sqrt {K[2]^2+1}+1\right )+a+a c_1{}^2}{2 \left (K[2]^2+1\right )}\right \},\{y(x),K[2]\}\right ] \\ y(x)\to 0 \\ \end{align*}