2.2.62 Problem 65
Internal
problem
ID
[13268]
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
:
65
Date
solved
:
Sunday, January 18, 2026 at 07:02:56 PM
CAS
classification
:
[_rational, _Riccati]
2.2.62.1 Solved using first_order_ode_riccati
1.772 (sec)
Entering first order ode riccati solver
\begin{align*}
x^{3} y^{\prime }&=a \,x^{3} y^{2}+\left (x^{2} b +c \right ) y+s x \\
\end{align*}
In canonical form the ODE is \begin{align*} y' &= F(x,y)\\ &= \frac {a \,x^{3} y^{2}+b \,x^{2} y+c y+s x}{x^{3}} \end{align*}
This is a Riccati ODE. Comparing the ODE to solve
\[
y' = a y^{2}+\frac {b y}{x}+\frac {c y}{x^{3}}+\frac {s}{x^{2}}
\]
With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]
Shows
that \(f_0(x)=\frac {s}{x^{2}}\), \(f_1(x)=\frac {b}{x}+\frac {c}{x^{3}}\) 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 &=\left (\frac {b}{x}+\frac {c}{x^{3}}\right ) a\\ f_2^2 f_0 &=\frac {a^{2} s}{x^{2}} \end{align*}
Substituting the above terms back in equation (2) gives
\[
a u^{\prime \prime }\left (x \right )-\left (\frac {b}{x}+\frac {c}{x^{3}}\right ) a u^{\prime }\left (x \right )+\frac {a^{2} s u \left (x \right )}{x^{2}} = 0
\]
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 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )+c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )
\end{equation}
Taking derivative gives \begin{equation}
\tag{4} u^{\prime }\left (x \right ) = \frac {c_1 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} \left (\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}\right ) {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x}+\frac {c_1 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} c \,{\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x^{3}}-\frac {2 c_1 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \left (\left (\frac {c}{2 x^{2}}+\frac {1}{4}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}\right ) \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )+\left (-\frac {1}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}-\frac {b}{4}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right )}{x}+\frac {c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} \left (\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}\right ) {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x}+\frac {c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} c \,{\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x^{3}}-\frac {2 c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \left (\left (\frac {c}{2 x^{2}}+\frac {1}{4}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}\right ) \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )-\operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right )}{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 {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} \left (\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}\right ) {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x}+\frac {c_1 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} c \,{\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x^{3}}-\frac {2 c_1 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \left (\left (\frac {c}{2 x^{2}}+\frac {1}{4}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}\right ) \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )+\left (-\frac {1}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}-\frac {b}{4}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right )}{x}+\frac {c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} \left (\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}\right ) {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x}+\frac {c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} c \,{\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x^{3}}-\frac {2 c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \left (\left (\frac {c}{2 x^{2}}+\frac {1}{4}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}\right ) \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )-\operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right )}{x}}{a \left (c_1 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )+c_2 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right )} \\
\end{align*}
Doing change of constants, the above solution becomes \[
y = -\frac {\frac {x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} \left (\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}\right ) {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x}+\frac {x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} c \,{\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x^{3}}-\frac {2 x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \left (\left (\frac {c}{2 x^{2}}+\frac {1}{4}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}\right ) \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )+\left (-\frac {1}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}-\frac {b}{4}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right )}{x}+\frac {c_3 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} \left (\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}\right ) {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x}+\frac {c_3 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} c \,{\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x^{3}}-\frac {2 c_3 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \left (\left (\frac {c}{2 x^{2}}+\frac {1}{4}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}\right ) \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )-\operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right )}{x}}{a \left (x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )+c_3 \,x^{\frac {1}{2}+\frac {b}{2}-\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}} {\mathrm e}^{-\frac {c}{2 x^{2}}} \operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right )}
\]
Simplifying the above gives
\begin{align*}
y &= -\frac {\left (c_3 \left (\sqrt {-4 a s +b^{2}+2 b +1}+b +1\right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )+\operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right ) a s \right ) \left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) x}{2 a \left (\frac {\left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) \left (\left (b -1\right ) x^{2}+c \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{2}+2 \left (\left (b -1\right ) x^{2}+c \right ) c_3 \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )+\left (\left (\frac {1}{2}+\frac {\left (b -1\right ) \sqrt {-4 a s +b^{2}+2 b +1}}{2}+a s -\frac {b^{2}}{2}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )-4 \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right ) c_3 \right ) x^{2}\right )} \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= -\frac {\left (c_3 \left (\sqrt {-4 a s +b^{2}+2 b +1}+b +1\right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )+\operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right ) a s \right ) \left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) x}{2 a \left (\frac {\left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) \left (\left (b -1\right ) x^{2}+c \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{2}+2 \left (\left (b -1\right ) x^{2}+c \right ) c_3 \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )+\left (\left (\frac {1}{2}+\frac {\left (b -1\right ) \sqrt {-4 a s +b^{2}+2 b +1}}{2}+a s -\frac {b^{2}}{2}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )-4 \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right ) c_3 \right ) x^{2}\right )} \\
\end{align*}
2.2.62.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 435
ode:=x^3*diff(y(x),x) = x^3*a*y(x)^2+(b*x^2+c)*y(x)+s*x;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\left (\sqrt {-4 a s +b^{2}+2 b +1}+b +1\right ) \left (\frac {\left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{4}+c_1 \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right ) x \left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right )}{2 \left (\frac {\left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) \left (\left (b -1\right ) x^{2}+c \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{2}+x^{2} \left (\frac {1}{2}+\frac {\left (b -1\right ) \sqrt {-4 a s +b^{2}+2 b +1}}{2}+a s -\frac {b^{2}}{2}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )-4 \left (\frac {\left (\left (-b +1\right ) x^{2}-c \right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{2}+\operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right ) x^{2}\right ) c_1 \right ) a}
\]
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) = (b*x^2+c)/x^3*diff(y
(x),x)-a/x^2*s*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
-> Kummer
-> hyper3: Equivalence to 1F1 under a power @ Moebius
<- hyper3 successful: received ODE is equivalent to the 1F1 ODE
<- Kummer successful
<- special function solution successful
<- Riccati to 2nd Order successful
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{3} \left (\frac {d}{d x}y \left (x \right )\right )=a \,x^{3} y \left (x \right )^{2}+\left (b \,x^{2}+c \right ) y \left (x \right )+s x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {a \,x^{3} y \left (x \right )^{2}+\left (b \,x^{2}+c \right ) y \left (x \right )+s x}{x^{3}} \end {array} \]
2.2.62.3 ✓ Mathematica. Time used: 1.281 (sec). Leaf size: 907
ode=x^3*D[y[x],x]==a*x^3*y[x]^2+(b*x^2+c)*y[x]+s*x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
2.2.62.4 ✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
s = symbols("s")
y = Function("y")
ode = Eq(-a*x**3*y(x)**2 - s*x + x**3*Derivative(y(x), x) - (b*x**2 + c)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a*y(x)**2 - b*y(x)/x - c*y(x)/x**3 - s/x**2 + Derivative(y(x),
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')