2.2.47 Problem 50

Internal problem ID [13253]
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 : 50
Date solved : Friday, December 19, 2025 at 02:08:08 AM
CAS classification : [_rational, _Riccati]

\begin{align*} y^{\prime } x^{2}&=c \,x^{2} y^{2}+\left (a \,x^{2}+b x \right ) y+\alpha \,x^{2}+\beta x +\gamma \\ \end{align*}

Entering first order ode riccati solver

\begin{align*} y^{\prime } x^{2}&=c \,x^{2} y^{2}+\left (a \,x^{2}+b x \right ) y+\alpha \,x^{2}+\beta x +\gamma \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= \frac {c \,x^{2} y^{2}+a \,x^{2} y +\alpha \,x^{2}+b x y +\beta x +\gamma }{x^{2}} \end{align*}

This is a Riccati ODE. Comparing the ODE to solve

\[ y' = y^{2} c +y a +\alpha +\frac {b y}{x}+\frac {\beta }{x}+\frac {\gamma }{x^{2}} \]

With Riccati ODE standard form

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

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{c u} \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 &=\frac {\left (a \,x^{2}+b x \right ) c}{x^{2}}\\ f_2^2 f_0 &=\frac {c^{2} \left (\alpha \,x^{2}+\beta x +\gamma \right )}{x^{2}} \end{align*}

Substituting the above terms back in equation (2) gives

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

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 \,{\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} \operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )+c_2 \,{\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} \operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) \end{equation}

Taking derivative gives

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

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{c u} \\ y &= -\frac {-c_1 \left (a b -2 c \beta -\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 c \alpha }-\sqrt {a^{2}-4 c \alpha }\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 c \beta -2 \sqrt {a^{2}-4 c \alpha }}{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 c \beta -2 \sqrt {a^{2}-4 c \alpha }}{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) c_2 \sqrt {a^{2}-4 c \alpha }+\left (\left (x a +b \right ) \sqrt {a^{2}-4 c \alpha }+a^{2} x +a b -4 \left (\alpha x +\frac {\beta }{2}\right ) c \right ) \left (\operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) c_2 +\operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) c_1 \right )}{2 \sqrt {a^{2}-4 c \alpha }\, x \left (\operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) c_2 +\operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) c_1 \right ) c} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {\frac {a \,{\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} \operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )}{2}+\frac {{\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} b \operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )}{2 x}+{\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} \left (\left (\frac {1}{2}+\frac {a b -2 c \beta }{2 \left (a^{2}-4 c \alpha \right ) x}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )+\frac {\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}+\frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}+\frac {1}{2}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}+1, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )}{\sqrt {a^{2}-4 c \alpha }\, x}\right ) \sqrt {a^{2}-4 c \alpha }+\frac {c_3 a \,{\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} \operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )}{2}+\frac {c_3 \,{\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} b \operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )}{2 x}+c_3 \,{\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} \left (\left (\frac {1}{2}+\frac {a b -2 c \beta }{2 \left (a^{2}-4 c \alpha \right ) x}\right ) \operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )-\frac {\operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}+1, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )}{\sqrt {a^{2}-4 c \alpha }\, x}\right ) \sqrt {a^{2}-4 c \alpha }}{c \left ({\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} \operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )+c_3 \,{\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} \operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )\right )} \]

Simplifying the above gives

\begin{align*} y &= -\frac {\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 c \alpha }-a b +2 c \beta +\sqrt {a^{2}-4 c \alpha }\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 c \beta -2 \sqrt {a^{2}-4 c \alpha }}{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 c \beta -2 \sqrt {a^{2}-4 c \alpha }}{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) c_3 \sqrt {a^{2}-4 c \alpha }+\left (\operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) c_3 +\operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )\right ) \left (\left (x a +b \right ) \sqrt {a^{2}-4 c \alpha }+a^{2} x +a b -4 \left (\alpha x +\frac {\beta }{2}\right ) c \right )}{2 \sqrt {a^{2}-4 c \alpha }\, x \left (\operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) c_3 +\operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )\right ) c} \\ \end{align*}

The solution

\[ y = -\frac {\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 c \alpha }-a b +2 c \beta +\sqrt {a^{2}-4 c \alpha }\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 c \beta -2 \sqrt {a^{2}-4 c \alpha }}{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 c \beta -2 \sqrt {a^{2}-4 c \alpha }}{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) c_3 \sqrt {a^{2}-4 c \alpha }+\left (\operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) c_3 +\operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )\right ) \left (\left (x a +b \right ) \sqrt {a^{2}-4 c \alpha }+a^{2} x +a b -4 \left (\alpha x +\frac {\beta }{2}\right ) c \right )}{2 \sqrt {a^{2}-4 c \alpha }\, x \left (\operatorname {WhittakerW}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right ) c_3 +\operatorname {WhittakerM}\left (-\frac {a b -2 c \beta }{2 \sqrt {a^{2}-4 c \alpha }}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 c \alpha }\, x \right )\right ) c} \]

was found not to satisfy the ode or the IC. Hence it is removed.

2.2.47.1 ✓ Maple. Time used: 0.004 (sec). Leaf size: 443

ode:=x^2*diff(y(x),x) = c*x^2*y(x)^2+(a*x^2+b*x)*y(x)+alpha*x^2+beta*x+gamma; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}-a b +2 \beta c +\sqrt {a^{2}-4 \alpha c}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_1 \sqrt {a^{2}-4 \alpha c}+\left (\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_1 +\operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) \left (\left (a x +b \right ) \sqrt {a^{2}-4 \alpha c}+a^{2} x +a b -4 c \left (\alpha x +\frac {\beta }{2}\right )\right )}{2 \sqrt {a^{2}-4 \alpha c}\, \left (\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_1 +\operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) c x} \]

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 sub-methods: 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = (a*x+b)/x*diff(y(x), 
x)-c*(alpha*x^2+beta*x+gamma)/x^2*y(x), y(x) 
      *** Sublevel 2 *** 
      Methods for second order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      checking if the LODE has constant coefficients 
      checking if the LODE is of Euler type 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> 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

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

2.2.47.2 ✓ Mathematica. Time used: 3.462 (sec). Leaf size: 1312

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

\begin{align*} \text {Solution too large to show}\end{align*}

2.2.47.3 ✗ Sympy

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

NotImplementedError : The given ODE -Alpha - BETA/x - Gamma/x**2 - a*y(x) - b*y(x)/x - c*y(x)**2 + D
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', 'lie_group')