60.2.4 problem 580

Internal problem ID [10591]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 580
Date solved : Monday, January 27, 2025 at 09:16:58 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} y^{\prime }&=F \left (y \,{\mathrm e}^{-b x}\right ) {\mathrm e}^{b x} \end{align*}

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 28

dsolve(diff(y(x),x) = F(y(x)*exp(-b*x))*exp(b*x),y(x), singsol=all)
 

\[ y = \operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )-b \textit {\_a}}d \textit {\_a} +c_{1} \right ) {\mathrm e}^{b x} \]

✓ Solution by Mathematica

Time used: 0.189 (sec). Leaf size: 203

DSolve[D[y[x],x] == E^(b*x)*F[y[x]/E^(b*x)],y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {1}{b K[2]-e^{b x} F\left (e^{-b x} K[2]\right )}-\int _1^x\left (\frac {F''\left (e^{-b K[1]} K[2]\right )}{e^{b K[1]} F\left (e^{-b K[1]} K[2]\right )-b K[2]}-\frac {e^{b K[1]} F\left (e^{-b K[1]} K[2]\right ) \left (F''\left (e^{-b K[1]} K[2]\right )-b\right )}{\left (e^{b K[1]} F\left (e^{-b K[1]} K[2]\right )-b K[2]\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {e^{b K[1]} F\left (e^{-b K[1]} y(x)\right )}{e^{b K[1]} F\left (e^{-b K[1]} y(x)\right )-b y(x)}dK[1]=c_1,y(x)\right ] \]