Internal problem ID [8828]
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')`]
\[ \boxed {\left (y^{2}-2 x a +a^{2}\right ) {y^{\prime }}^{2}+2 a y y^{\prime }+y^{2}=0} \]
✓ Solution by Maple
Time used: 0.453 (sec). Leaf size: 113
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 {\sqrt {\textit {\_T}^{2}+1}\, {\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )}^{2} a^{2}+\left (-2 a c_{1} \sqrt {\textit {\_T}^{2}+1}-2 a^{2}\right ) \operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )+\left (a^{2}+c_{1}^{2}\right ) \sqrt {\textit {\_T}^{2}+1}+2 c_{1} a}{2 \sqrt {\textit {\_T}^{2}+1}\, a}, 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: 49.544 (sec). Leaf size: 408
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 \text {arctanh}\left (\sqrt {K[2]^2+1}\right )^2+a \text {arctanh}\left (\sqrt {K[2]^2+1}\right )^2-2 a \sqrt {K[2]^2+1} \text {arctanh}\left (\sqrt {K[2]^2+1}\right )+2 a c_1 K[2]^2 \text {arctanh}\left (\sqrt {K[2]^2+1}\right )+2 a c_1 \text {arctanh}\left (\sqrt {K[2]^2+1}\right )+a K[2]^2+a c_1{}^2 K[2]^2-2 a c_1 \sqrt {K[2]^2+1}+a+a c_1{}^2}{2 \left (K[2]^2+1\right )}\right \},\{y(x),K[2]\}\right ] \\ y(x)\to 0 \\ \end{align*}