2.2.5 Problem 5

Internal problem ID [13211]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number : 5
Date solved : Wednesday, December 31, 2025 at 12:03:06 PM
CAS classification : [_Riccati]

2.2.5.1 Solved using first_order_ode_riccati_by_guessing_particular_solution

26.707 (sec)

Entering first order ode riccati guess solver

\begin{align*} y^{\prime }&=y^{2}+a n \,x^{n -1}-a^{2} x^{2 n} \\ \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_1(x) & =0\\ f_2(x) &=1 \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

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

Summary of solutions found

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

2.2.5.2 ✓ Maple. Time used: 0.002 (sec). Leaf size: 366

ode:=diff(y(x),x) = y(x)^2+a*n*x^(n-1)-a^2*x^(2*n); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-3 \left (n +2\right ) c_1 \left (\left (\frac {1}{3} n^{2}+n +\frac {2}{3}\right ) x^{-\frac {3 n}{2}}+a x \,x^{-\frac {n}{2}} \left (n +\frac {4}{3}\right )\right ) \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {3+2 n}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )+2 \left (\left (-\frac {1}{2} n^{2}-\frac {3}{2} n -1\right ) x^{-\frac {3 n}{2}}+a x \left (\left (-\frac {n}{2}-1\right ) x^{-\frac {n}{2}}+a x \,x^{\frac {n}{2}}\right )\right ) \left (n +1\right ) c_1 \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {3+2 n}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )+2 \left (n +\frac {3}{2}\right ) \left (n +2\right )^{2} {\mathrm e}^{\frac {a \,x^{n} x}{n +1}} c_1 \,x^{-\frac {3 n}{2}} \left (-\frac {2 a \,x^{n} x}{n +1}\right )^{\frac {4+3 n}{2 n +2}}+2 a \,x^{n} x^{2} {\mathrm e}^{-\frac {a \,x^{n} x}{n +1}}}{2 x \left (-\frac {x^{-\frac {3 n}{2}} c_1 \left (n +2\right )^{2} \operatorname {WhittakerM}\left (\frac {n +2}{2 n +2}, \frac {3+2 n}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )}{2}+\left (\left (-\frac {n}{2}-1\right ) x^{-\frac {3 n}{2}}+x^{-\frac {n}{2}} a x \right ) \left (n +1\right ) c_1 \operatorname {WhittakerM}\left (-\frac {n}{2 n +2}, \frac {3+2 n}{2 n +2}, -\frac {2 a \,x^{n} x}{n +1}\right )+{\mathrm e}^{-\frac {a \,x^{n} x}{n +1}} x \right )} \]

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 Special 
trying Riccati sub-methods: 
   trying Riccati_symmetries 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = (-a*n*x^(n-1)+a^2*x^ 
(2*n))*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 Liouvillian solution using Kovacics algorithm 
            A Liouvillian solution exists 
            Reducible group (found an exponential solution) 
            Group is reducible, not completely reducible 
         <- Kovacics algorithm successful 
      <- Equivalence, under non-integer power transformations successful 
   <- Riccati to 2nd Order successful
 

2.2.5.3 ✓ Mathematica. Time used: 0.686 (sec). Leaf size: 227

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

\begin{align*} y(x)&\to \frac {2^{\frac {1}{n+1}} (n+1) \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}} \left (a x^n-c_1 e^{\frac {2 a x^{n+1}}{n+1}}\right )-a c_1 x^{n+1} \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )}{2^{\frac {1}{n+1}} (n+1) \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}}-c_1 x \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )}\\ y(x)&\to \frac {2^{\frac {1}{n+1}} (n+1) e^{\frac {2 a x^{n+1}}{n+1}} \left (-\frac {a x^{n+1}}{n+1}\right )^{\frac {1}{n+1}}}{x \Gamma \left (\frac {1}{n+1},-\frac {2 a x^{n+1}}{n+1}\right )}+a x^n \end{align*}

2.2.5.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a**2*x**(2*n) - a*n*x**(n - 1) - 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) - a*n*x**(n - 1) - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method