Internal problem ID [8160]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 580.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime }-F \left (y \,{\mathrm e}^{-x b}\right ) {\mathrm e}^{x b}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 31
dsolve(diff(y(x),x) = F(y(x)*exp(-b*x))*exp(b*x),y(x), singsol=all)
\[ y \relax (x ) = \RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )-b \textit {\_a}}d \textit {\_a} +c_{1}\right ) {\mathrm e}^{b x} \]
✓ Solution by Mathematica
Time used: 0.197 (sec). Leaf size: 203
DSolve[y'[x] == E^(b*x)*F[y[x]/E^(b*x)],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{b K[2]-e^{b x} F\left (e^{-b x} K[2]\right )}-\int _1^x\left (\frac {F'\left (e^{-b K[1]} K[2]\right )}{e^{b K[1]} F\left (e^{-b K[1]} K[2]\right )-b K[2]}-\frac {e^{b K[1]} F\left (e^{-b K[1]} K[2]\right ) \left (F'\left (e^{-b K[1]} K[2]\right )-b\right )}{\left (e^{b K[1]} F\left (e^{-b K[1]} K[2]\right )-b K[2]\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {e^{b K[1]} F\left (e^{-b K[1]} y(x)\right )}{e^{b K[1]} F\left (e^{-b K[1]} y(x)\right )-b y(x)}dK[1]=c_1,y(x)\right ] \]