2.493   ODE No. 493

\[ \left (a^2-2 a x+y(x)^2\right ) y'(x)^2+2 a y(x) y'(x)+y(x)^2=0 \]

✓ Mathematica : cpu = 14.289 (sec), leaf count = 97

DSolve[y[x]^2 + 2*a*y[x]*Derivative[1][y][x] + (a^2 - 2*a*x + y[x]^2)*Derivative[1][y][x]^2 == 0,y[x],x]
 

\[\text {Solve}\left [\left \{x=\frac {a^2 K[1]^2+2 a K[1] y(K[1])+K[1]^2 y(K[1])^2+y(K[1])^2}{2 a K[1]^2},y(x)=-\frac {a K[1] \tanh ^{-1}\left (\sqrt {K[1]^2+1}\right )}{\sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}}\right \},\{y(x),K[1]\}\right ]\]

✓ Maple : cpu = 1.013 (sec), leaf count = 109

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))
 

\[\left [x \left (\textit {\_T} \right ) = \frac {\sqrt {\textit {\_T}^{2}+1}\, {\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )}^{2} a^{2}+\left (-2 a c_{1} \sqrt {\textit {\_T}^{2}+1}-2 a^{2}\right ) \operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )+\left (a^{2}+c_{1}^{2}\right ) \sqrt {\textit {\_T}^{2}+1}+2 c_{1} a}{2 \sqrt {\textit {\_T}^{2}+1}\, a}, y \left (\textit {\_T} \right ) = \frac {\left (-a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )+c_{1} \right ) \textit {\_T}}{\sqrt {\textit {\_T}^{2}+1}}\right ]\]