2.4.4 Problem 25

Internal problem ID [13304]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3-2. Equations with power and exponential functions
Problem number : 25
Date solved : Sunday, January 18, 2026 at 07:10:58 PM
CAS classification : [_Riccati]

2.4.4.1 Solved using first_order_ode_riccati_by_guessing_particular_solution

0.099 (sec)

Entering first order ode riccati guess solver

\begin{align*} y^{\prime }&={\mathrm e}^{\lambda x} y^{2}+a \,x^{n} y+a \lambda \,x^{n} {\mathrm e}^{-\lambda x} \\ \end{align*}

This is a Riccati ODE. Comparing the above ODE to solve with the Riccati standard form

\[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]

Shows that

\begin{align*} f_0(x) & =a \lambda \,x^{n} {\mathrm e}^{-\lambda x}\\ f_1(x) & =x^{n} a\\ f_2(x) &={\mathrm e}^{\lambda x} \end{align*}

Using trial and error, the following particular solution was found

\[ y_p = -\lambda \,{\mathrm e}^{-\lambda x} \]

Since a particular solution is known, then the general solution is given by

\begin{align*} y &= y_p + \frac { \phi (x) }{ c_1 - \int { \phi (x) f_2 \,dx} } \end{align*}

Where

\begin{align*} \phi (x) &= e^{ \int 2 f_2 y_p + f_1 \,dx } \end{align*}

Evaluating the above gives the general solution as

\[ y = -\lambda \,{\mathrm e}^{-\lambda x}+\frac {{\mathrm e}^{\frac {a \,x^{n +1}}{n +1}-2 \lambda x}}{c_1 -\int {\mathrm e}^{\frac {a \,x^{n +1}}{n +1}-2 \lambda x} {\mathrm e}^{\lambda x}d x} \]

Summary of solutions found

\begin{align*} y &= -\lambda \,{\mathrm e}^{-\lambda x}+\frac {{\mathrm e}^{\frac {a \,x^{n +1}}{n +1}-2 \lambda x}}{c_1 -\int {\mathrm e}^{\frac {a \,x^{n +1}}{n +1}-2 \lambda x} {\mathrm e}^{\lambda x}d x} \\ \end{align*}

2.4.4.2 ✓ Maple. Time used: 0.004 (sec). Leaf size: 88

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

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

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
Looking for potential symmetries 
trying Riccati 
trying Riccati sub-methods: 
   trying Riccati_symmetries 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = (a*x^n+lambda)*diff( 
y(x),x)-exp(lambda*x)*a*lambda*x^n*exp(-lambda*x)*y(x), y(x) 
      *** Sublevel 2 *** 
      Methods for second order ODEs: 
      --- Trying classification methods --- 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> Trying an equivalence, under non-integer power transformations, 
         to LODEs admitting Liouvillian solutions. 
      -> Trying a solution in terms of special functions: 
         -> Bessel 
         -> elliptic 
         -> Legendre 
         -> Kummer 
            -> hyper3: Equivalence to 1F1 under a power @ Moebius 
         -> hypergeometric 
            -> heuristic approach 
            -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
         -> Mathieu 
            -> Equivalence to the rational form of Mathieu ODE under a power @\ 
 Moebius 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \ 
power @ Moebius 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \ 
power @ Moebius 
      -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int\ 
(r(x), dx)) * 2F1([a1, a2], [b1], f) 
      -> Trying changes of variables to rationalize or make the ODE simpler 
      <- unable to find a useful change of variables 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         trying 2nd order exact linear 
         trying symmetries linear in x and y(x) 
         trying to convert to a linear ODE with constant coefficients 
         trying 2nd order, integrating factor of the form mu(x,y) 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         checking if the LODE is missing y 
         -> Trying an equivalence, under non-integer power transformations, 
            to LODEs admitting Liouvillian solutions. 
         -> Trying a solution in terms of special functions: 
            -> Bessel 
            -> elliptic 
            -> Legendre 
            -> Kummer 
               -> hyper3: Equivalence to 1F1 under a power @ Moebius 
            -> hypergeometric 
               -> heuristic approach 
               -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebi\ 
us 
            -> Mathieu 
               -> Equivalence to the rational form of Mathieu ODE under a powe\ 
r @ Moebius 
         -> Heun: Equivalence to the GHE or one of its 4 confluent cases under \ 
a power @ Moebius 
         -> Heun: Equivalence to the GHE or one of its 4 confluent cases under \ 
a power @ Moebius 
         -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(\ 
int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
         -> Trying changes of variables to rationalize or make the ODE simpler 
         <- unable to find a useful change of variables 
            trying a symmetry of the form [xi=0, eta=F(x)] 
         trying to convert to an ODE of Bessel type 
   -> Trying a change of variables to reduce to Bernoulli 
   -> Calling odsolve with the ODE, diff(y(x),x)-(exp(lambda*x)*y(x)^2+y(x)+a*x 
^n*y(x)*x+x^2*a*lambda*x^n*exp(-lambda*x))/x, y(x), explicit 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      trying Bernoulli 
      trying separable 
      trying inverse linear 
      trying homogeneous types: 
      trying Chini 
      differential order: 1; looking for linear symmetries 
      trying exact 
      Looking for potential symmetries 
      trying Riccati 
      trying Riccati sub-methods: 
         trying Riccati_symmetries 
      trying inverse_Riccati 
      trying 1st order ODE linearizable_by_differentiation 
   -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
   -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
   -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
trying inverse_Riccati 
trying 1st order ODE linearizable_by_differentiation 
--- Trying Lie symmetry methods, 1st order --- 
   -> Computing symmetries using: way = 4 
   -> Computing symmetries using: way = 2 
   -> Computing symmetries using: way = 6 
[0, exp(-a*x^(n+1)/(n+1)+2*lambda*x)*(y+lambda/exp(lambda*x))^2] 
   <- successful computation of symmetries. 
1st order, trying the canonical coordinates of the invariance group 
<- 1st order, canonical coordinates successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )={\mathrm e}^{\lambda x} y \left (x \right )^{2}+a \,x^{13304} y \left (x \right )+a \lambda \,x^{13304} {\mathrm e}^{-\lambda x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )={\mathrm e}^{\lambda x} y \left (x \right )^{2}+a \,x^{13304} y \left (x \right )+a \lambda \,x^{13304} {\mathrm e}^{-\lambda x} \end {array} \]

2.4.4.3 ✓ Mathematica. Time used: 0.578 (sec). Leaf size: 254

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

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

2.4.4.4 ✗ Sympy

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

NotImplementedError : The given ODE -a*lambda_*x**n*exp(-lambda_*x) - a*x**n*y(x) - y(x)**2*exp(lamb
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('1st_power_series', 'lie_group')