Problem 6

2.2.6 Problem 6

2.2.6.1 Solved using ﬁrst_order_ode_riccati
2.2.6.2 ✓Maple
2.2.6.3 ✓Mathematica
2.2.6.4 ✗Sympy

Internal problem ID [13212]
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 : 6
Date solved : Wednesday, December 31, 2025 at 12:04:22 PM
CAS classiﬁcation : [_Riccati]

2.2.6.1 Solved using ﬁrst_order_ode_riccati

10.518 (sec)

Entering ﬁrst order ode riccati solver

\begin{align*} y^{\prime }&=a y^{2}+b \,x^{2 n}+c \,x^{n -1} \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= a y^{2}+b \,x^{2 n}+c \,x^{n -1} \end{align*}

This is a Riccati ODE. Comparing the ODE to solve

\[ y' = \textit {the\_rhs} \]

With Riccati ODE standard form

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

Shows that \(f_0(x)=b \,x^{2 n}+\frac {c \,x^{n}}{x}\), \(f_1(x)=0\) and \(f_2(x)=a\). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u a} \tag {1} \end{align*}

Using the above substitution in the given ODE results (after some simpliﬁcation) in a second order ODE to solve for \(u(x)\) which is

\begin{align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end{align*}

But

\begin{align*} f_2' &=0\\ f_1 f_2 &=0\\ f_2^2 f_0 &=a^{2} \left (b \,x^{2 n}+\frac {c \,x^{n}}{x}\right ) \end{align*}

Substituting the above terms back in equation (2) gives

\[ a u^{\prime \prime }\left (x \right )+a^{2} \left (b \,x^{2 n}+\frac {c \,x^{n}}{x}\right ) u \left (x \right ) = 0 \]

Unable to solve. Will ask Maple to solve this ode now.

The solution for \(u \left (x \right )\) is

\begin{equation} \tag{3} u \left (x \right ) = c_1 \,x^{-\frac {n}{2}} \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+c_2 \,x^{-\frac {n}{2}} \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right ) \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = -\frac {c_1 \,x^{-\frac {n}{2}} n \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 x}+\frac {2 i c_1 \,x^{-\frac {n}{2}} \left (\left (\frac {1}{2}+\frac {c \left (n +1\right ) x^{-n -1}}{2 b \left (2 n +2\right )}\right ) \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )-\frac {i \left (\frac {1}{2}+\frac {1}{2 n +2}-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}\right ) \left (n +1\right ) x^{-n -1} \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}+1, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, x^{n +1}}{x}-\frac {c_2 \,x^{-\frac {n}{2}} n \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 x}+\frac {2 i c_2 \,x^{-\frac {n}{2}} \left (\left (\frac {1}{2}+\frac {c \left (n +1\right ) x^{-n -1}}{2 b \left (2 n +2\right )}\right ) \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+\frac {i \left (n +1\right ) x^{-n -1} \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}+1, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, x^{n +1}}{x} \end{equation}

Substituting equations (3,4) into (1) results in

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u a} \\ y &= -\frac {-\frac {c_1 \,x^{-\frac {n}{2}} n \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 x}+\frac {2 i c_1 \,x^{-\frac {n}{2}} \left (\left (\frac {1}{2}+\frac {c \left (n +1\right ) x^{-n -1}}{2 b \left (2 n +2\right )}\right ) \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )-\frac {i \left (\frac {1}{2}+\frac {1}{2 n +2}-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}\right ) \left (n +1\right ) x^{-n -1} \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}+1, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, x^{n +1}}{x}-\frac {c_2 \,x^{-\frac {n}{2}} n \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 x}+\frac {2 i c_2 \,x^{-\frac {n}{2}} \left (\left (\frac {1}{2}+\frac {c \left (n +1\right ) x^{-n -1}}{2 b \left (2 n +2\right )}\right ) \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+\frac {i \left (n +1\right ) x^{-n -1} \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}+1, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, x^{n +1}}{x}}{a \left (c_1 \,x^{-\frac {n}{2}} \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+c_2 \,x^{-\frac {n}{2}} \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )\right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {-\frac {x^{-\frac {n}{2}} n \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 x}+\frac {2 i x^{-\frac {n}{2}} \left (\left (\frac {1}{2}+\frac {c \left (n +1\right ) x^{-n -1}}{2 b \left (2 n +2\right )}\right ) \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )-\frac {i \left (\frac {1}{2}+\frac {1}{2 n +2}-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}\right ) \left (n +1\right ) x^{-n -1} \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}+1, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, x^{n +1}}{x}-\frac {c_3 \,x^{-\frac {n}{2}} n \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 x}+\frac {2 i c_3 \,x^{-\frac {n}{2}} \left (\left (\frac {1}{2}+\frac {c \left (n +1\right ) x^{-n -1}}{2 b \left (2 n +2\right )}\right ) \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+\frac {i \left (n +1\right ) x^{-n -1} \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}+1, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )}{2 \sqrt {a}\, \sqrt {b}}\right ) \sqrt {a}\, \sqrt {b}\, x^{n +1}}{x}}{a \left (x^{-\frac {n}{2}} \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+c_3 \,x^{-\frac {n}{2}} \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )\right )} \]

Simplifying the above gives

\begin{align*} y &= \frac {\left (x^{-n -1} x^{\frac {n}{2}} \left (i \sqrt {a}\, b c -\left (n +2\right ) b^{{3}/{2}}\right ) x^{n} x \operatorname {WhittakerM}\left (1-\frac {i \sqrt {a}\, c}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+2 x^{-n -1} c_3 \,b^{{3}/{2}} x \,x^{n} x^{\frac {n}{2}} \left (n +1\right ) \operatorname {WhittakerW}\left (1-\frac {i \sqrt {a}\, c}{2 \sqrt {b}\, \left (n +1\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )+n \,x^{-1-\frac {n}{2}} \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right ) b^{{3}/{2}} x^{n} x +n c_3 \,x^{-1-\frac {n}{2}} \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right ) b^{{3}/{2}} x^{n} x -2 i \sqrt {a}\, b \,x^{\frac {n}{2}} \left (x^{n} b x +\frac {c}{2}\right ) \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n} x}{n +1}\right ) c_3 +\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n} x}{n +1}\right )\right )\right ) x^{\frac {n}{2}} x^{-n}}{2 b^{{3}/{2}} a \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right ) c_3 +\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n +1}}{n +1}\right )\right ) x} \\ \end{align*}

Summary of solutions found

2.2.6.2 ✓ Maple. Time used: 0.002 (sec). Leaf size: 357

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

\[ y = -\frac {\left (\left (1+\frac {n}{2}\right ) \sqrt {b}-\frac {i c \sqrt {a}}{2}\right ) \operatorname {WhittakerM}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i c \sqrt {a}}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n} x}{n +1}\right )-\sqrt {b}\, c_1 \left (n +1\right ) \operatorname {WhittakerW}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i c \sqrt {a}}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n} x}{n +1}\right )+\left (-\frac {\sqrt {b}\, n}{2}+i \left (x^{n} b x +\frac {c}{2}\right ) \sqrt {a}\right ) \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n} x}{n +1}\right ) c_1 +\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n} x}{n +1}\right )\right )}{\sqrt {b}\, \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n} x}{n +1}\right ) c_1 +\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x^{n} x}{n +1}\right )\right ) a x} \]

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 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*(b*x^(2*n)+c*x^(n 
-1))*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 
         <- No Liouvillian solutions exists 
      -> Trying a solution in terms of special functions: 
         -> Bessel 
         -> elliptic 
         -> Legendre 
         -> Whittaker 
            -> hyper3: Equivalence to 1F1 under a power @ Moebius 
            <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
         <- Whittaker successful 
      <- special function solution successful 
   <- Riccati to 2nd Order successful

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a y \left (x \right )^{2}+b \,x^{26424}+c \,x^{13211} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=a y \left (x \right )^{2}+b \,x^{26424}+c \,x^{13211} \end {array} \]

2.2.6.3 ✓ Mathematica. Time used: 0.732 (sec). Leaf size: 764

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

\begin{align*} y(x)&\to -\frac {x^n \left (\sqrt {b} c_1 (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \left (\sqrt {a} c (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {3 n+2}{n+1}\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\sqrt {b} (n+1) \sqrt {-(n+1)^2} \left (L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+2 L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {3 n+2}{2 n+2}}^{\frac {n}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )\right )}{\sqrt {a} (n+1)^2 \left (L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )}\\ y(x)&\to \frac {x^n \left (-\frac {\left (\sqrt {a} c (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}+2\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}-\sqrt {b} \sqrt {-(n+1)^2} (n+1)\right )}{\sqrt {a} (n+1)^2} \end{align*}

2.2.6.4 ✗ Sympy

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

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

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