60.2.79 problem 655

Internal problem ID [10666]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 655
Date solved : Monday, January 27, 2025 at 09:23:17 PM
CAS classification : [[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {y^{3} {\mathrm e}^{-\frac {4 x}{3}}}{y \,{\mathrm e}^{-\frac {2 x}{3}}+1} \end{align*}

✓ Solution by Maple

Time used: 1.182 (sec). Leaf size: 64

dsolve(diff(y(x),x) = y(x)^3/(y(x)*exp(-2/3*x)+1)*exp(-4/3*x),y(x), singsol=all)
 

\[ x +\frac {3 \sqrt {7}\, \operatorname {arctanh}\left (\frac {3 y \sqrt {7}\, {\mathrm e}^{-\frac {2 x}{3}}}{7}-\frac {\sqrt {7}}{7}\right )}{14}-\frac {3 \ln \left (3 y^{2} {\mathrm e}^{-\frac {4 x}{3}}-2 y \,{\mathrm e}^{-\frac {2 x}{3}}-2\right )}{4}+\frac {3 \ln \left (y \,{\mathrm e}^{-\frac {2 x}{3}}\right )}{2}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 9.074 (sec). Leaf size: 123

DSolve[D[y[x],x] == y[x]^3/(E^((4*x)/3)*(1 + y[x]/E^((2*x)/3))),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {-11 e^{4 x/3} y(x)-2 e^{2 x}}{2^{2/3} \sqrt [3]{31} \sqrt [3]{-e^{4 x}} \left (y(x)+e^{2 x/3}\right )}}\frac {1}{K[1]^3+\frac {33 \sqrt [3]{-\frac {1}{2}} K[1]}{31^{2/3}}+1}dK[1]=\frac {2}{81} \sqrt [3]{2} 31^{2/3} e^{-8 x/3} \left (-e^{4 x}\right )^{2/3} x+c_1,y(x)\right ] \]