Internal problem ID [3625]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 30
Problem number: 897.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {x^{2} \left (y^{\prime }\right )^{2}+2 a x y^{\prime }+a^{2}+x^{2}-2 a y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.091 (sec). Leaf size: 236
dsolve(x^2*diff(y(x),x)^2+2*a*x*diff(y(x),x)+a^2+x^2-2*a*y(x) = 0,y(x), singsol=all)
\[ y \relax (x ) = \frac {\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right )^{2} x^{2}}{2 a}-\frac {a \arcsinh \left (\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right )\right ) \RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right )}{\sqrt {\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right )^{2}+1}}+\frac {a}{2}+\frac {x^{2}}{2 a}+\frac {c_{1} \RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right )}{\sqrt {\RootOf \left (\arcsinh \left (\textit {\_Z} \right )^{2} a^{2}-x^{2} \textit {\_Z}^{2}-2 \arcsinh \left (\textit {\_Z} \right ) c_{1} a +c_{1}^{2}-x^{2}\right )^{2}+1}} \]
✓ Solution by Mathematica
Time used: 1.024 (sec). Leaf size: 81
DSolve[x^2 (y'[x])^2+2 a x y'[x]+a^2+x^2-2 a y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{y(x)=\frac {2 a x K[1]+x^2 K[1]^2+a^2+x^2}{2 a},x=-\frac {a \tanh ^{-1}\left (\frac {K[1]}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1}}+\frac {c_1}{\sqrt {K[1]^2+1}}\right \},\{y(x),K[1]\}\right ] \]