Internal problem ID [8174]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 594.
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 {-y^{2}+b}{x^{2}}\right ) x}{y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 95
dsolve(diff(y(x),x) = F(-(-y(x)^2+b)/x^2)*x/y(x),y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (-F \left (\frac {\textit {\_Z}^{2}-b}{x^{2}}\right ) x^{2}+\textit {\_Z}^{2}-b \right ) \\ y \relax (x ) = \sqrt {\RootOf \left (-2 \ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )-\textit {\_a}}d \textit {\_a} +2 c_{1}\right ) x^{2}+b} \\ y \relax (x ) = -\sqrt {\RootOf \left (-2 \ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )-\textit {\_a}}d \textit {\_a} +2 c_{1}\right ) x^{2}+b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.249 (sec). Leaf size: 236
DSolve[y'[x] == (x*F[(-b + y[x]^2)/x^2])/y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]}{-F\left (\frac {K[2]^2-b}{x^2}\right ) x^2+K[2]^2-b}-\int _1^x\left (\frac {F\left (\frac {K[2]^2-b}{K[1]^2}\right ) K[1] \left (2 K[2] F'\left (\frac {K[2]^2-b}{K[1]^2}\right )-2 K[2]\right )}{\left (F\left (\frac {K[2]^2-b}{K[1]^2}\right ) K[1]^2-K[2]^2+b\right )^2}-\frac {2 K[2] F'\left (\frac {K[2]^2-b}{K[1]^2}\right )}{K[1] \left (F\left (\frac {K[2]^2-b}{K[1]^2}\right ) K[1]^2-K[2]^2+b\right )}\right )dK[1]\right )dK[2]+\int _1^x-\frac {F\left (\frac {y(x)^2-b}{K[1]^2}\right ) K[1]}{F\left (\frac {y(x)^2-b}{K[1]^2}\right ) K[1]^2-y(x)^2+b}dK[1]=c_1,y(x)\right ] \]