60.2.122 problem 698

Internal problem ID [10709]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 698
Date solved : Monday, January 27, 2025 at 09:28:49 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Abel]

\begin{align*} y^{\prime }&=\left (1+y^{2} {\mathrm e}^{-2 x}+y^{3} {\mathrm e}^{-3 x}\right ) {\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 30

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

\[ y = \operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a}^{3}+\textit {\_a}^{2}-\textit {\_a} +1}d \textit {\_a} +c_{1} \right ) {\mathrm e}^{x} \]

✓ Solution by Mathematica

Time used: 0.275 (sec). Leaf size: 89

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

\[ \text {Solve}\left [\int _1^{\frac {3 e^{-2 x} y(x)+e^{-x}}{\sqrt [3]{38} \sqrt [3]{e^{-3 x}}}}\frac {1}{K[1]^3-\frac {6 \sqrt [3]{2} K[1]}{19^{2/3}}+1}dK[1]=\frac {1}{9} 38^{2/3} e^{2 x} \left (e^{-3 x}\right )^{2/3} x+c_1,y(x)\right ] \]