Internal problem ID [8768]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 432.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational]
\[ \boxed {\left (y^{\prime } x +a \right )^{2}-2 y a=-x^{2}} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 86
dsolve((x*diff(y(x),x)+a)^2-2*a*y(x)+x^2 = 0,y(x), singsol=all)
\[ y \left (x \right )-\operatorname {RootOf}\left (-a \,\operatorname {arcsinh}\left (\frac {\operatorname {RootOf}\left (-2 a y \left (x \right )+a^{2}+x^{2}+2 a \textit {\_Z} +\textit {\_Z}^{2}\right )}{x}\right )-x \sqrt {-\frac {2 a \operatorname {RootOf}\left (-2 a y \left (x \right )+a^{2}+x^{2}+2 a \textit {\_Z} +\textit {\_Z}^{2}\right )}{x^{2}}-\frac {a^{2}}{x^{2}}+\frac {2 a \textit {\_Z}}{x^{2}}}+c_{1} \right ) = 0 \]
✓ Solution by Mathematica
Time used: 0.895 (sec). Leaf size: 82
DSolve[x^2 - 2*a*y[x] + (a + x*y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{y(x)=\frac {2 a x K[1]+x^2 K[1]^2+a^2+x^2}{2 a},x=\frac {a \log \left (\sqrt {K[1]^2+1}-K[1]\right )}{\sqrt {K[1]^2+1}}+\frac {c_1}{\sqrt {K[1]^2+1}}\right \},\{y(x),K[1]\}\right ] \]