2.19.5 Problem 5

Internal problem ID [13453]
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 : 5
Date solved : Wednesday, December 31, 2025 at 09:39:18 PM
CAS classification : [_Riccati]

2.19.5.1 Solved using first_order_ode_riccati_by_guessing_particular_solution

0.245 (sec)

Entering first order ode riccati guess solver

\begin{align*} y^{\prime }&=f \left (x \right ) y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} 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) & =\frac {a n \,x^{n}}{x}-a^{2} x^{2 n} f \left (x \right )\\ f_1(x) & =0\\ f_2(x) &=f \left (x \right ) \end{align*}

Using trial and error, the following particular solution was found

\[ y_p = x^{n} a \]

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

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

Summary of solutions found

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

2.19.5.2 ✗ Maple

ode:=diff(y(x),x) = f(x)*y(x)^2+a*n*x^(n-1)-a^2*x^(2*n)*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)*(a*x^(2*n)*diff(f(x),x)*x 
+2*a*x^(2*n)*n*f(x)-x^(n-1)*n^2+x^(n-1)*n)/x/(f(x)*x^(2*n)*a-x^(n-1)*n), 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

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=f \left (x \right ) y \left (x \right )^{2}+13453 a \,x^{13452}-a^{2} x^{26906} f \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=f \left (x \right ) y \left (x \right )^{2}+13453 a \,x^{13452}-a^{2} x^{26906} f \left (x \right ) \end {array} \]

2.19.5.3 ✗ Mathematica

ode=D[y[x],x]==f[x]*y[x]^2+a*n*x^(n-1)-a^2*x^(2*n)*f[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.19.5.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
f = Function("f") 
ode = Eq(a**2*x**(2*n)*f(x) - a*n*x**(n - 1) - f(x)*y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a**2*x**(2*n)*f(x) - a*n*x**(n - 1) - f(x)*y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method