55.3.16 problem 16

Internal problem ID [13320]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3. Equations Containing Exponential Functions
Problem number : 16
Date solved : Wednesday, October 01, 2025 at 07:06:58 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a \,{\mathrm e}^{k x} y^{2}+b y+c \,{\mathrm e}^{s x}+d \,{\mathrm e}^{-k x} \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 334

ode:=diff(y(x),x) = a*exp(k*x)*y(x)^2+b*y(x)+c*exp(s*x)+d*exp(-k*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\frac {x \left (k -s \right )}{2}} \sqrt {c}\, a \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}+k +s}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right ) c_1 +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}+k +s}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right )\right )-\frac {{\mathrm e}^{-k x} \sqrt {a}\, \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right ) c_1 +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right )\right ) \left (\sqrt {-4 a d +b^{2}+2 b k +k^{2}}+b +k \right )}{2}}{a^{{3}/{2}} \left (\operatorname {BesselY}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right ) c_1 +\operatorname {BesselJ}\left (\frac {\sqrt {-4 a d +b^{2}+2 b k +k^{2}}}{k +s}, \frac {2 \sqrt {c}\, \sqrt {a}\, {\mathrm e}^{\frac {x \left (k +s \right )}{2}}}{k +s}\right )\right )} \]

✓ Mathematica. Time used: 5.127 (sec). Leaf size: 1636

ode=D[y[x],x]==a*Exp[k*x]*y[x]^2+b*y[x]+c*Exp[s*x]+d*Exp[-k*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -a*y(x)**2*exp(k*x) - b*y(x) - c*exp(s*x) - d*exp(-k*x) + Derivative(y(x), x) cannot be solved by the lie group method