81.7.6 problem 8-5
Internal
problem
ID
[21580]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
I.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
8.
Riccati
Equation.
Page
124.
Problem
number
:
8-5
Date
solved
:
Thursday, October 02, 2025 at 07:55:47 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&={\mathrm e}^{2 x}+\left (2+\frac {5 \,{\mathrm e}^{x}}{2}\right ) y+y^{2} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 157
ode:=diff(y(x),x) = exp(2*x)+(2+5/2*exp(x))*y(x)+y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (-1440 \,{\mathrm e}^{x}-5400 \,{\mathrm e}^{2 x}+4860 \,{\mathrm e}^{3 x}-320\right ) \operatorname {KummerM}\left (\frac {1}{3}, 3, \frac {3 \,{\mathrm e}^{x}}{2}\right )-1296 c_1 \left ({\mathrm e}^{x}+\frac {15 \,{\mathrm e}^{2 x}}{4}-\frac {27 \,{\mathrm e}^{3 x}}{8}+\frac {2}{9}\right ) \operatorname {KummerU}\left (\frac {1}{3}, 3, \frac {3 \,{\mathrm e}^{x}}{2}\right )-1296 \left ({\mathrm e}^{x}+\frac {9 \,{\mathrm e}^{2 x}}{4}+\frac {1}{3}\right ) \left (c_1 \operatorname {KummerU}\left (-\frac {2}{3}, 3, \frac {3 \,{\mathrm e}^{x}}{2}\right )-\frac {80 \operatorname {KummerM}\left (-\frac {2}{3}, 3, \frac {3 \,{\mathrm e}^{x}}{2}\right )}{27}\right )}{\left (4320 \,{\mathrm e}^{x}-2430 \,{\mathrm e}^{2 x}-1440\right ) \operatorname {KummerM}\left (\frac {1}{3}, 3, \frac {3 \,{\mathrm e}^{x}}{2}\right )+3888 c_1 \left ({\mathrm e}^{x}-\frac {9 \,{\mathrm e}^{2 x}}{16}-\frac {1}{3}\right ) \operatorname {KummerU}\left (\frac {1}{3}, 3, \frac {3 \,{\mathrm e}^{x}}{2}\right )+1458 \left (-\frac {2}{9}+{\mathrm e}^{x}\right ) \left (c_1 \operatorname {KummerU}\left (-\frac {2}{3}, 3, \frac {3 \,{\mathrm e}^{x}}{2}\right )-\frac {80 \operatorname {KummerM}\left (-\frac {2}{3}, 3, \frac {3 \,{\mathrm e}^{x}}{2}\right )}{27}\right )}
\]
✓ Mathematica. Time used: 0.518 (sec). Leaf size: 172
ode=D[y[x],x]==Exp[2*x]+(2+5/2*Exp[x])*y[x]+y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {c_1 \left (e^x+4\right ) \operatorname {HypergeometricU}\left (\frac {7}{3},3,\frac {3 e^x}{2}\right )-7 c_1 e^x \operatorname {HypergeometricU}\left (\frac {10}{3},4,\frac {3 e^x}{2}\right )-3 e^x L_{-\frac {10}{3}}^3\left (\frac {3 e^x}{2}\right )+e^x L_{-\frac {7}{3}}^2\left (\frac {3 e^x}{2}\right )+4 L_{-\frac {7}{3}}^2\left (\frac {3 e^x}{2}\right )}{2 \left (L_{-\frac {7}{3}}^2\left (\frac {3 e^x}{2}\right )+c_1 \operatorname {HypergeometricU}\left (\frac {7}{3},3,\frac {3 e^x}{2}\right )\right )}\\ y(x)&\to \frac {7 e^x \operatorname {HypergeometricU}\left (\frac {10}{3},4,\frac {3 e^x}{2}\right )}{2 \operatorname {HypergeometricU}\left (\frac {7}{3},3,\frac {3 e^x}{2}\right )}-\frac {e^x}{2}-2 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-5*exp(x)/2 - 2)*y(x) - y(x)**2 - exp(2*x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -y(x)**2 - 5*y(x)*exp(x)/2 - 2*y(x) - exp(2*x) + Derivative(y(x), x) cannot be solved by the factorable group method