60.1.86 problem 88

Internal problem ID [10100]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 88
Date solved : Wednesday, March 05, 2025 at 08:30:54 AM
CAS classification : [_Riccati]

\begin{align*} 2 y^{\prime }-3 y^{2}-4 a y-b -c \,{\mathrm e}^{-2 a x}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 256

ode:=2*diff(y(x),x)-3*y(x)^2-4*a*y(x)-b-c*exp(-2*a*x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-{\mathrm e}^{-a x} \left (\operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}-2 a}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}-2 a}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )\right ) \sqrt {3}\, \sqrt {c}-\left (\sqrt {4 a^{2}-3 b}+2 a \right ) \left (\operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )\right )}{3 \operatorname {BesselY}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right ) c_{1} +3 \operatorname {BesselJ}\left (-\frac {\sqrt {4 a^{2}-3 b}}{2 a}, \frac {\sqrt {3}\, \sqrt {c}\, {\mathrm e}^{-a x}}{2 a}\right )} \]

✓ Mathematica. Time used: 1.848 (sec). Leaf size: 2746

ode=2*D[y[x],x] - 3*y[x]^2 - 4*a*y[x] - b - c*Exp[-2*a*x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-4*a*y(x) - b - c*exp(-2*a*x) - 3*y(x)**2 + 2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -2*a*y(x) - b/2 - c*exp(-2*a*x)/2 - 3*y(x)**2/2 + Derivative(y(x), x) cannot be solved by the factorable group method