Internal problem ID [8187]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 607.
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 {2 y+F \left (\frac {y}{x^{2}}\right ) x^{3}}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 32
dsolve(diff(y(x),x) = (2*y(x)+F(1/x^2*y(x))*x^3)/x,y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (F \left (\textit {\_Z} \right )\right ) x^{2} \\ y \relax (x ) = \RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1}\right ) x^{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.184 (sec). Leaf size: 121
DSolve[y'[x] == (x^3*F[y[x]/x^2] + 2*y[x])/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {F\left (\frac {K[2]}{x^2}\right ) \int _1^x\left (\frac {2}{F\left (\frac {K[2]}{K[1]^2}\right ) K[1]^3}-\frac {2 K[2] F'\left (\frac {K[2]}{K[1]^2}\right )}{F\left (\frac {K[2]}{K[1]^2}\right )^2 K[1]^5}\right )dK[1] x^2+1}{x^2 F\left (\frac {K[2]}{x^2}\right )}dK[2]+\int _1^x\left (\frac {2 y(x)}{F\left (\frac {y(x)}{K[1]^2}\right ) K[1]^3}+1\right )dK[1]=c_1,y(x)\right ] \]