60.2.57 problem 633

Internal problem ID [10644]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 633
Date solved : Monday, January 27, 2025 at 09:20:16 PM
CAS classification : [[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {{\mathrm e}^{\frac {2 x}{3}}}{y \,{\mathrm e}^{-\frac {2 x}{3}}+1} \end{align*}

✓ Solution by Maple

Time used: 1.178 (sec). Leaf size: 49

dsolve(diff(y(x),x) = 1/(y(x)*exp(-2/3*x)+1)*exp(2/3*x),y(x), singsol=all)
 

\[ y = \operatorname {RootOf}\left (-{\mathrm e}^{\operatorname {RootOf}\left (343 \operatorname {sech}\left (\frac {\left (4 c_{1} -4 x -3 \textit {\_Z} \right ) \sqrt {7}}{6}\right )^{2}+98 \,{\mathrm e}^{\textit {\_Z}}\right )}-3+2 \textit {\_Z} +2 \textit {\_Z}^{2}\right ) {\mathrm e}^{\frac {2 x}{3}} \]

✓ Solution by Mathematica

Time used: 0.167 (sec). Leaf size: 85

DSolve[D[y[x],x] == E^((2*x)/3)/(1 + y[x]/E^((2*x)/3)),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [7 \left (3 \log \left (-\frac {2}{3} e^{-4 x/3} y(x)^2-\frac {2}{3} e^{-2 x/3} y(x)+1\right )+4 x-9 c_1\right )=6 \sqrt {7} \text {arctanh}\left (\frac {y(x)+4 e^{2 x/3}}{\sqrt {7} \left (y(x)+e^{2 x/3}\right )}\right ),y(x)\right ] \]