55.3.21 problem 21

Internal problem ID [13325]
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 : 21
Date solved : Wednesday, October 01, 2025 at 07:21:49 AM
CAS classification : [_Riccati]

\begin{align*} \left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} {\mathrm e}^{\lambda x}+b \,\mu ^{2} {\mathrm e}^{\mu x}&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 176

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

\[ y = \frac {\left (-a b \left (\lambda +\mu \right ) {\mathrm e}^{x \left (\lambda +\mu \right )}-a^{2} \lambda \,{\mathrm e}^{2 \lambda x}-b^{2} \mu \,{\mathrm e}^{2 \mu x}-c \left (a \lambda \,{\mathrm e}^{\lambda x}+b \mu \,{\mathrm e}^{\mu x}\right )\right ) \int \frac {1}{\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x -a b c_1 \left (\lambda +\mu \right ) {\mathrm e}^{x \left (\lambda +\mu \right )}-a^{2} \lambda \,{\mathrm e}^{2 \lambda x} c_1 -{\mathrm e}^{\lambda x} c_1 a c \lambda -b^{2} \mu \,{\mathrm e}^{2 \mu x} c_1 -{\mathrm e}^{\mu x} c_1 b c \mu -1}{\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right )^{2} \left (c_1 +\int \frac {1}{\left (a \,{\mathrm e}^{\lambda x}+b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right )} \]

✓ Mathematica. Time used: 5.528 (sec). Leaf size: 393

ode=(a*Exp[\[Lambda]*x]+b*Exp[\[Mu]*x]+c)*(D[y[x],x]-y[x]^2)+a*\[Lambda]^2*Exp[\[Lambda]*x]+b*\[Mu]^2*Exp[\[Mu]*x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

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