54.2.394 problem 973

Internal problem ID [12268]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 973
Date solved : Wednesday, October 01, 2025 at 01:26:30 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Abel]

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

✓ Maple. Time used: 0.186 (sec). Leaf size: 166

ode:=diff(y(x),x) = y(x)*(y(x)^2+y(x)*exp(b*x)+exp(b*x)^2)/exp(b*x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{b x}}{2}-\frac {\tan \left (\operatorname {RootOf}\left (-2 \sqrt {-{\mathrm e}^{2 b x} \left (4 b -3\right )}\, {\mathrm e}^{-b x} b x +\sqrt {-{\mathrm e}^{2 b x} \left (4 b -3\right )}\, \ln \left (\frac {{\mathrm e}^{2 b x} \left (\tan \left (\textit {\_Z} \right )^{2}+1\right ) \left (4 b -3\right )}{\left (\tan \left (\textit {\_Z} \right ) \sqrt {-{\mathrm e}^{2 b x} \left (4 b -3\right )}+{\mathrm e}^{b x}\right )^{2}}\right ) {\mathrm e}^{-b x}+\sqrt {-{\mathrm e}^{2 b x} \left (4 b -3\right )}\, {\mathrm e}^{-b x} c_1 +2 \sqrt {-{\mathrm e}^{2 b x} \left (4 b -3\right )}\, {\mathrm e}^{-b x} x -2 \textit {\_Z} \right )\right ) \sqrt {{\mathrm e}^{2 b x} \left (-4 b +3\right )}}{2} \]

✓ Mathematica. Time used: 0.194 (sec). Leaf size: 100

ode=D[y[x],x] == (y[x]*(E^(2*b*x) + E^(b*x)*y[x] + y[x]^2))/E^(2*b*x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {3 e^{-2 b x} y(x)+e^{-b x}}{\sqrt [3]{(9 b-7) e^{-3 b x}}}}\frac {1}{K[1]^3-\frac {3 (3 b-2) K[1]}{(9 b-7)^{2/3}}+1}dK[1]=\frac {1}{9} x e^{2 b x} \left ((9 b-7) e^{-3 b x}\right )^{2/3}+c_1,y(x)\right ] \]

✓ Sympy. Time used: 16.307 (sec). Leaf size: 743

from sympy import * 
x = symbols("x") 
b = symbols("b") 
y = Function("y") 
ode = Eq((-y(x)**2 - y(x)*exp(b*x) - exp(2*b*x))*y(x)*exp(-2*b*x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]