Internal problem ID [8895]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 560.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_rational]
\[ \boxed {a y \sqrt {{y^{\prime }}^{2}+1}-2 y x y^{\prime }+y^{2}=x^{2}} \]
✓ Solution by Maple
Time used: 1.421 (sec). Leaf size: 1086
dsolve(a*y(x)*(diff(y(x),x)^2+1)^(1/2)-2*x*y(x)*diff(y(x),x)+y(x)^2-x^2=0,y(x), singsol=all)
\begin{align*} -\left (\int _{\textit {\_b}}^{x}\frac {-2 y \left (x \right )^{2} \textit {\_a} +2 \textit {\_a}^{3}+\sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}}{\left (-\textit {\_a}^{2}+y \left (x \right )^{2}\right ) \sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}-2 \textit {\_a} y \left (x \right )^{4}+\left (-4 \textit {\_a}^{3}+2 \textit {\_a} \,a^{2}\right ) y \left (x \right )^{2}-2 \textit {\_a}^{5}}d \textit {\_a} \right )+\int _{}^{y \left (x \right )}\frac {4 \textit {\_f} \left (\left (\frac {\left (\textit {\_f}^{2}-x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}}{2}-x^{5}-2 x^{3} \textit {\_f}^{2}+\left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x \right ) \left (\int _{\textit {\_b}}^{x}\frac {\left (\left (-4 \textit {\_a}^{2}-a^{2}\right ) \textit {\_f}^{4}+\left (8 \textit {\_a}^{4}-2 \textit {\_a}^{2} a^{2}+a^{4}\right ) \textit {\_f}^{2}-5 \textit {\_a}^{4} a^{2}+12 \textit {\_a}^{6}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}+a^{2} \textit {\_a} \left (-2 \textit {\_a}^{2}-2 \textit {\_f}^{2}+a^{2}\right ) \left (-2 \textit {\_a}^{4}-2 \textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right )}{\sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\, \left (2 \textit {\_a} \,\textit {\_f}^{2} a^{2}-2 \textit {\_a}^{5}-4 \textit {\_a}^{3} \textit {\_f}^{2}-2 \textit {\_a} \,\textit {\_f}^{4}-\textit {\_a}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}+\textit {\_f}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\right )^{2}}d \textit {\_a} \right )+\frac {a^{2}}{4}-x^{2}\right )}{\left (\textit {\_f}^{2}-x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}-2 x^{5}-4 x^{3} \textit {\_f}^{2}+2 \left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x}d \textit {\_f} +c_{1} &= 0 \\ -\left (\int _{\textit {\_b}}^{x}\frac {2 y \left (x \right )^{2} \textit {\_a} -2 \textit {\_a}^{3}+\sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}}{2 \textit {\_a} y \left (x \right )^{4}+4 \textit {\_a}^{3} y \left (x \right )^{2}-2 \textit {\_a} y \left (x \right )^{2} a^{2}+2 \textit {\_a}^{5}+y \left (x \right )^{2} \sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}-\textit {\_a}^{2} \sqrt {a^{2} \left (y \left (x \right )^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) y \left (x \right )^{2}+\textit {\_a}^{4}\right )}}d \textit {\_a} \right )+\int _{}^{y \left (x \right )}\frac {4 \textit {\_f} \left (\left (\frac {\left (-\textit {\_f}^{2}+x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}}{2}-x^{5}-2 x^{3} \textit {\_f}^{2}+\left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x \right ) \left (\int _{\textit {\_b}}^{x}-\frac {\left (\left (4 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{4}+\left (-8 \textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}-a^{4}\right ) \textit {\_f}^{2}+5 \textit {\_a}^{4} a^{2}-12 \textit {\_a}^{6}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}+a^{2} \textit {\_a} \left (-2 \textit {\_a}^{2}-2 \textit {\_f}^{2}+a^{2}\right ) \left (-2 \textit {\_a}^{4}-2 \textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right )}{\sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\, \left (2 \textit {\_a} \,\textit {\_f}^{2} a^{2}-2 \textit {\_a}^{5}-4 \textit {\_a}^{3} \textit {\_f}^{2}-2 \textit {\_a} \,\textit {\_f}^{4}+\textit {\_a}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}-\textit {\_f}^{2} \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (-2 \textit {\_a}^{2}+a^{2}\right ) \textit {\_f}^{2}+\textit {\_a}^{4}\right )}\right )^{2}}d \textit {\_a} \right )+\frac {a^{2}}{4}-x^{2}\right )}{\left (-\textit {\_f}^{2}+x^{2}\right ) \sqrt {a^{2} \left (\textit {\_f}^{4}-\left (a^{2}-2 x^{2}\right ) \textit {\_f}^{2}+x^{4}\right )}-2 x^{5}-4 x^{3} \textit {\_f}^{2}+2 \left (-\textit {\_f}^{4}+\textit {\_f}^{2} a^{2}\right ) x}d \textit {\_f} +c_{1} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 57.481 (sec). Leaf size: 135
DSolve[-x^2 + y[x]^2 - 2*x*y[x]*y'[x] + a*y[x]*Sqrt[1 + y'[x]^2]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {4 x^2-a^2 (2+c_1 x){}^2}}{\sqrt {-4+a^2 c_1{}^2}} \\ y(x)\to \frac {\sqrt {4 x^2-a^2 (2+c_1 x){}^2}}{\sqrt {-4+a^2 c_1{}^2}} \\ y(x)\to -\frac {\sqrt {-a^2 x^2}}{\sqrt {a^2}} \\ y(x)\to \frac {\sqrt {-a^2 x^2}}{\sqrt {a^2}} \\ \end{align*}