2.19.14 Problem 14

Internal problem ID [13462]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.8-1. Equations containing arbitrary functions (but not containing their derivatives).
Problem number : 14
Date solved : Wednesday, December 31, 2025 at 09:42:03 PM
CAS classification : [_Riccati]

2.19.14.1 Solved using first_order_ode_riccati_by_guessing_particular_solution

0.247 (sec)

Entering first order ode riccati guess solver

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

Using trial and error, the following particular solution was found

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

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

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

Summary of solutions found

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

2.19.14.2 ✗ Maple

ode:=diff(y(x),x) = f(x)*y(x)^2+a*lambda*exp(lambda*x)-a^2*exp(2*lambda*x)*f(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 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 
trying symmetry patterns for 1st order ODEs 
-> 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 symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 
   -> Computing symmetries using: way = HINT 
   -> Calling odsolve with the ODE, diff(y(x),x) = 0, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x) = 2*y(x)/x, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)+diff(f(x),x)*y(x)/f(x), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)+y(x)*(diff(f(x),x)*exp(2* 
lambda*x)*a+2*f(x)*lambda*exp(2*lambda*x)*a-lambda^2*exp(lambda*x))/(f(x)*exp(2 
*lambda*x)*a-lambda*exp(lambda*x)), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x) = -diff(f(x),x)*y(x)/f(x), y(x 
) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x), y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Calling odsolve with the ODE, diff(y(x),x)-2*y(x)/x, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
-> trying a symmetry pattern of the form [F(x),G(x)] 
-> trying a symmetry pattern of the form [F(y),G(y)] 
-> 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 a symmetry pattern of conformal type
 

Maple step by step

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

2.19.14.3 ✗ Mathematica

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

Not solved

2.19.14.4 ✗ Sympy

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

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