Internal problem ID [8193]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 613.
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 {x +y+F \left (-\frac {-y+x \ln \relax (x )}{x}\right ) x^{2}}{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.028 (sec). Leaf size: 34
dsolve(diff(y(x),x) = (x+y(x)+F(-(-y(x)+x*ln(x))/x)*x^2)/x,y(x), singsol=all)
\begin{align*} y \relax (x ) = x \left (\RootOf \left (F \left (\textit {\_Z} \right )\right )+\ln \relax (x )\right ) \\ y \relax (x ) = \left (\ln \relax (x )+\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1}\right )\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.219 (sec). Leaf size: 226
DSolve[y'[x] == (x + x^2*F[(-(x*Log[x]) + y[x])/x] + y[x])/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x F\left (\frac {K[2]-x \log (x)}{x}\right ) \int _1^x\left (-\frac {K[2] F'\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )}{F\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )^2 K[1]^3}-\frac {F'\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )}{F\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right )^2 K[1]^2}+\frac {1}{F\left (\frac {K[2]-K[1] \log (K[1])}{K[1]}\right ) K[1]^2}\right )dK[1]+1}{x F\left (\frac {K[2]-x \log (x)}{x}\right )}dK[2]+\int _1^x\left (\frac {y(x)}{F\left (\frac {y(x)-K[1] \log (K[1])}{K[1]}\right ) K[1]^2}+\frac {1}{F\left (\frac {y(x)-K[1] \log (K[1])}{K[1]}\right ) K[1]}+1\right )dK[1]=c_1,y(x)\right ] \]