Internal problem ID [8884]
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: 6.
CAS Maple gives this as type [_quadrature]
\[ \boxed {x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}=a^{2}} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 601
dsolve(x^2*(diff(y(x),x)^2+1)^3-a^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {-\sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}\, a^{2}+c_{1} \left (a^{2} x \right )^{\frac {2}{3}}+\left (a^{2} x \right )^{\frac {2}{3}} \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}}{\left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right ) \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (a^{2}-\left (a^{2} x \right )^{\frac {2}{3}}\right )}{a^{2}}}+c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{\left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}-\sqrt {3}\, a^{2}\right )}{a^{2}}}\, \left (2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}+i \sqrt {3}\, a^{2}\right )}{4 \sqrt {\left (i \left (a^{2} x \right )^{\frac {1}{3}}+2 i x -\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}\right ) x}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}+\left (a^{2} x \right )^{\frac {1}{3}}+2 x \right )}\, \sqrt {\frac {x \left (a^{2} x \right )^{\frac {1}{3}} \left (2 i \left (a^{2} x \right )^{\frac {2}{3}}+i a^{2}-\sqrt {3}\, a^{2}\right )}{a^{2}}}\, \left (2 \left (a^{2} x \right )^{\frac {2}{3}}+a^{2}+i \sqrt {3}\, a^{2}\right )}{4 \sqrt {\left (i \left (a^{2} x \right )^{\frac {1}{3}}+2 i x -\sqrt {3}\, \left (a^{2} x \right )^{\frac {1}{3}}\right ) x}\, \left (a^{2} x \right )^{\frac {2}{3}}}+c_{1} \\ y \left (x \right ) &= \frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}\right ) \sqrt {2}\, \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}\right ) x \left (a^{2} x \right )^{\frac {1}{3}}}{a^{2}}}+4 c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{4 \left (a^{2} x \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (-2 \left (a^{2} x \right )^{\frac {2}{3}} \sqrt {2}+a^{2} \left (i \sqrt {6}-\sqrt {2}\right )\right ) \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{\frac {2}{3}}\right ) x \left (a^{2} x \right )^{\frac {1}{3}}}{a^{2}}}-4 c_{1} \left (a^{2} x \right )^{\frac {2}{3}}}{4 \left (a^{2} x \right )^{\frac {2}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 19.706 (sec). Leaf size: 375
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}{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 )+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}{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 )+c_1 \\ \end{align*}