Internal problem ID [8227]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 647.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\left (a y^{2}+x^{2} b \right )^{2} x}{a^{\frac {5}{2}} y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 460
dsolve(diff(y(x),x) = (a*y(x)^2+b*x^2)^2*x/a^(5/2)/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {-a \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right ) \left (\left (b \,x^{2}-a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\right ) {\mathrm e}^{\frac {x^{2} \left (-2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+c_{1} \left (a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right ) {\mathrm e}^{\frac {x^{2} \left (2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right )}}{a \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right )} \\ y \relax (x ) = -\frac {\sqrt {-a \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right ) \left (\left (b \,x^{2}-a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}\right ) {\mathrm e}^{\frac {x^{2} \left (-2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+c_{1} \left (a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right ) {\mathrm e}^{\frac {x^{2} \left (2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right )}}{a \left (c_{1} {\mathrm e}^{\frac {x^{2} \left (2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}+{\mathrm e}^{\frac {x^{2} \left (-2 a^{\frac {3}{2}} \sqrt {-\frac {b}{a^{\frac {3}{2}}}}+b \,x^{2}\right )}{2 a^{\frac {3}{2}}}}\right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 11.643 (sec). Leaf size: 117
DSolve[y'[x] == (x*(b*x^2 + a*y[x]^2)^2)/(a^(5/2)*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {\frac {-b x^2+a^{3/4} \sqrt {b} \tan \left (\frac {a^{3/2} b x^2+2 c_1}{a^{9/4} \sqrt {b}}\right )}{a}} \\ y(x)\to \sqrt {\frac {-b x^2+a^{3/4} \sqrt {b} \tan \left (\frac {a^{3/2} b x^2+2 c_1}{a^{9/4} \sqrt {b}}\right )}{a}} \\ \end{align*}