61.3.20 problem 20

Internal problem ID [12025]
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 : 20
Date solved : Friday, March 14, 2025 at 03:48:41 AM
CAS classification : [_Riccati]

\begin{align*} \left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) y^{\prime }&=y^{2}+k \,{\mathrm e}^{\nu x} y-m^{2}+k m \,{\mathrm e}^{\nu x} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 202

ode:=(exp(lambda*x)*a+b*exp(x*mu)+c)*diff(y(x),x) = y(x)^2+k*exp(nu*x)*y(x)-m^2+k*m*exp(nu*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-m \left (\int \frac {{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}d x \right )}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}d x \right )+c_{1} m -{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}d x \right )}}{\int \frac {{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}d x \right )}}{{\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c}d x -c_{1}} \]

✓ Mathematica. Time used: 5.56 (sec). Leaf size: 358

ode=(a*Exp[\[Lambda]*x]+b*Exp[\[Mu]*x]+c)*D[y[x],x]==y[x]^2+k*Exp[\[Nu]*x]*y[x]-m^2+k*m*Exp[\[Nu]*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[2]}-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right ) \left (e^{\nu K[2]} k-m+y(x)\right )}{\left (e^{\lambda K[2]} a+b e^{\mu K[2]}+c\right ) k \nu (m+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right )}{k \nu (m+K[3])^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right ) \left (e^{\nu K[2]} k-m+K[3]\right )}{\left (e^{\lambda K[2]} a+b e^{\mu K[2]}+c\right ) k \nu (m+K[3])^2}-\frac {\exp \left (-\int _1^{K[2]}-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right )}{\left (e^{\lambda K[2]} a+b e^{\mu K[2]}+c\right ) k \nu (m+K[3])}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out