2.4.7 Problem 28

Internal problem ID [13307]
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 : 28
Date solved : Wednesday, December 31, 2025 at 01:10:01 PM
CAS classification : [_Riccati]

2.4.7.1 Solved using first_order_ode_riccati_by_guessing_particular_solution

0.155 (sec)

Entering first order ode riccati guess solver

\begin{align*} y^{\prime }&=a \,x^{n} y^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \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) & =b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x}\\ f_1(x) & =0\\ f_2(x) &=x^{n} a \end{align*}

Using trial and error, the following particular solution was found

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

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

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

Summary of solutions found

\begin{align*} y &= b \,{\mathrm e}^{\lambda x}+\frac {{\mathrm e}^{-\frac {2 \left (-\lambda \right )^{-n} a b \left (x^{n} \left (-\lambda \right )^{n} n \Gamma \left (n \right ) \left (-\lambda x \right )^{-n}-x^{n} \left (-\lambda \right )^{n} {\mathrm e}^{\lambda x}-x^{n} \left (-\lambda \right )^{n} n \left (-\lambda x \right )^{-n} \Gamma \left (n , -\lambda x \right )\right )}{\lambda }}}{c_1 -\int {\mathrm e}^{-\frac {2 \left (-\lambda \right )^{-n} a b \left (x^{n} \left (-\lambda \right )^{n} n \Gamma \left (n \right ) \left (-\lambda x \right )^{-n}-x^{n} \left (-\lambda \right )^{n} {\mathrm e}^{\lambda x}-x^{n} \left (-\lambda \right )^{n} n \left (-\lambda x \right )^{-n} \Gamma \left (n , -\lambda x \right )\right )}{\lambda }} x^{n} a d x} \\ \end{align*}

2.4.7.2 ✗ Maple

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

\[ \text {No solution found} \]

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) = n/x*diff(y(x),x)+a*x 
^n*b*(x^n*exp(2*lambda*x)*a*b-lambda*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 
      -> 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 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         checking if the LODE is missing y 
         -> 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 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 
      <- 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 
         -> 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 
            trying a symmetry of the form [xi=0, eta=F(x)] 
            checking if the LODE is missing y 
            -> Heun: Equivalence to the GHE or one of its 4 confluent cases und\ 
er a power @ Moebius 
            -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = e\ 
xp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
               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 
         <- 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)-(a*x^n*y(x)^2+y(x)+x^2*(b* 
lambda*exp(lambda*x)-a*b^2*x^n*exp(2*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
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \,x^{13307} y \left (x \right )^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{13307} {\mathrm e}^{2 \lambda x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a \,x^{13307} y \left (x \right )^{2}+b \lambda \,{\mathrm e}^{\lambda x}-a \,b^{2} x^{13307} {\mathrm e}^{2 \lambda x} \end {array} \]

2.4.7.3 ✗ Mathematica

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

Not solved

2.4.7.4 ✗ Sympy

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

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