Internal problem ID [8129]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 549.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {x^{2} \left (\left (y^{\prime }\right )^{2}+1\right )^{3}-a^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.202 (sec). Leaf size: 552
dsolve(x^2*(diff(y(x),x)^2+1)^3-a^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \relax (x ) = -\frac {\sqrt {-\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{a^{4}}}\, \left (\left (a^{2} x \right )^{\frac {2}{3}}-a^{2}\right )}{\left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \relax (x ) = -\frac {i \sqrt {2}\, \sqrt {-i \left (\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-i \left (a^{2} x \right )^{\frac {1}{3}}-2 i x \right ) x}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\sqrt {3}\, a^{2}-2 i \left (a^{2} x \right )^{\frac {2}{3}}-i a^{2}\right )}{a^{4}}}\, \left (\sqrt {3}\, a^{2}-2 i \left (a^{2} x \right )^{\frac {2}{3}}-i a^{2}\right )}{4 \sqrt {\left (\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-i \left (a^{2} x \right )^{\frac {1}{3}}-2 i x \right ) x}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \relax (x ) = \frac {i \sqrt {2}\, \sqrt {-i \left (\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-i \left (a^{2} x \right )^{\frac {1}{3}}-2 i x \right ) x}\, \sqrt {\frac {\left (a^{2} x \right )^{\frac {4}{3}} \left (\sqrt {3}\, a^{2}-2 i \left (a^{2} x \right )^{\frac {2}{3}}-i a^{2}\right )}{a^{4}}}\, \left (\sqrt {3}\, a^{2}-2 i \left (a^{2} x \right )^{\frac {2}{3}}-i a^{2}\right )}{4 \sqrt {\left (\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}-i \left (a^{2} x \right )^{\frac {1}{3}}-2 i x \right ) x}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \relax (x ) = \frac {i \sqrt {2}\, \sqrt {\frac {i \left (a^{2} x \right )^{\frac {4}{3}} \left (\sqrt {3}\, a^{2}+2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}\right )}{a^{4}}}\, \left (\sqrt {3}\, a^{2}+2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \relax (x ) = -\frac {i \sqrt {2}\, \sqrt {\frac {i \left (a^{2} x \right )^{\frac {4}{3}} \left (\sqrt {3}\, a^{2}+2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}\right )}{a^{4}}}\, \left (\sqrt {3}\, a^{2}+2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}\right )}{4 \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 19.544 (sec). Leaf size: 319
DSolve[-a^2 + x^2*(1 + y'[x]^2)^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt [3]{x} \sqrt {\frac {a^{2/3}}{x^{2/3}}-1} \left (x^{2/3}-a^{2/3}\right )+c_1 \\ y(x)\to \sqrt [3]{x} \sqrt {\frac {a^{2/3}}{x^{2/3}}-1} \left (a^{2/3}-x^{2/3}\right )+c_1 \\ y(x)\to c_1-\frac {1}{2} \sqrt [3]{x} \sqrt {-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}} \left (2 x^{2/3}+\left (1-i \sqrt {3}\right ) a^{2/3}\right ) \\ y(x)\to \frac {1}{4} \sqrt [3]{x} \sqrt {-4+\frac {2 i \left (\sqrt {3}+i\right ) a^{2/3}}{x^{2/3}}} \left (2 x^{2/3}+\left (1-i \sqrt {3}\right ) a^{2/3}\right )+c_1 \\ y(x)\to \frac {x \left (-2+\frac {\left (-1-i \sqrt {3}\right ) a^{2/3}}{x^{2/3}}\right )^{3/2}}{2 \sqrt {2}}+c_1 \\ y(x)\to c_1-\frac {x \left (-2+\frac {\left (-1-i \sqrt {3}\right ) a^{2/3}}{x^{2/3}}\right )^{3/2}}{2 \sqrt {2}} \\ \end{align*}