Internal problem ID [8171]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 591.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {F \left (\frac {a y^{2}+x^{2} b}{a}\right ) x}{\sqrt {a}\, y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 171
dsolve(diff(y(x),x) = F((a*y(x)^2+b*x^2)/a)*x/a^(1/2)/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\sqrt {a \left (-b \,x^{2}+a \RootOf \left (F \left (\textit {\_Z} \right ) a +b \sqrt {a}\right )\right )}}{a} \\ y \relax (x ) = -\frac {\sqrt {a \left (-b \,x^{2}+a \RootOf \left (F \left (\textit {\_Z} \right ) a +b \sqrt {a}\right )\right )}}{a} \\ y \relax (x ) = \frac {\sqrt {a \left (-b \,x^{2}+\RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right ) a +b \sqrt {a}}d \textit {\_a} \right ) b \,a^{\frac {3}{2}}-b \,x^{2}+2 a c_{1}\right ) a \right )}}{a} \\ y \relax (x ) = -\frac {\sqrt {a \left (-b \,x^{2}+\RootOf \left (\left (\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right ) a +b \sqrt {a}}d \textit {\_a} \right ) b \,a^{\frac {3}{2}}-b \,x^{2}+2 a c_{1}\right ) a \right )}}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.365 (sec). Leaf size: 253
DSolve[y'[x] == (x*F[(b*x^2 + a*y[x]^2)/a])/(Sqrt[a]*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {b K[2]}{b+\sqrt {a} F\left (\frac {b x^2+a K[2]^2}{a}\right )}-\int _1^x\left (\frac {2 b K[1] K[2] F'\left (\frac {b K[1]^2+a K[2]^2}{a}\right )}{\sqrt {a} \left (b+\sqrt {a} F\left (\frac {b K[1]^2+a K[2]^2}{a}\right )\right )}-\frac {2 b F\left (\frac {b K[1]^2+a K[2]^2}{a}\right ) K[1] K[2] F'\left (\frac {b K[1]^2+a K[2]^2}{a}\right )}{\left (b+\sqrt {a} F\left (\frac {b K[1]^2+a K[2]^2}{a}\right )\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {b F\left (\frac {b K[1]^2+a y(x)^2}{a}\right ) K[1]}{\sqrt {a} \left (b+\sqrt {a} F\left (\frac {b K[1]^2+a y(x)^2}{a}\right )\right )}dK[1]=c_1,y(x)\right ] \]