2.2.64 Problem 67

Internal problem ID [13270]
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 : 67
Date solved : Wednesday, December 31, 2025 at 12:46:25 PM
CAS classification : [_rational, _Riccati]

2.2.64.1 Solved using first_order_ode_riccati

16.119 (sec)

Entering first order ode riccati solver

\begin{align*} x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y+s x&=0 \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= -\frac {y^{2} \lambda \,x^{3}+y^{2} a \lambda x +b \,x^{2} y+c y+s x}{x \left (x^{2}+a \right )} \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 \(f_0(x)=-\frac {s}{x^{2}+a}\), \(f_1(x)=-\frac {x b}{x^{2}+a}-\frac {c}{x \left (x^{2}+a \right )}\) and \(f_2(x)=-\frac {x^{2} \lambda }{x^{2}+a}-\frac {a \lambda }{x^{2}+a}\). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u \left (-\frac {x^{2} \lambda }{x^{2}+a}-\frac {a \lambda }{x^{2}+a}\right )} \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' &=-\frac {2 x \lambda }{x^{2}+a}+\frac {2 x^{3} \lambda }{\left (x^{2}+a \right )^{2}}+\frac {2 a \lambda x}{\left (x^{2}+a \right )^{2}}\\ f_1 f_2 &=\left (-\frac {x b}{x^{2}+a}-\frac {c}{x \left (x^{2}+a \right )}\right ) \left (-\frac {x^{2} \lambda }{x^{2}+a}-\frac {a \lambda }{x^{2}+a}\right )\\ f_2^2 f_0 &=-\frac {\left (-\frac {x^{2} \lambda }{x^{2}+a}-\frac {a \lambda }{x^{2}+a}\right )^{2} s}{x^{2}+a} \end{align*}

Substituting the above terms back in equation (2) gives

\[ \left (-\frac {x^{2} \lambda }{x^{2}+a}-\frac {a \lambda }{x^{2}+a}\right ) u^{\prime \prime }\left (x \right )-\left (-\frac {2 x \lambda }{x^{2}+a}+\frac {2 x^{3} \lambda }{\left (x^{2}+a \right )^{2}}+\frac {2 a \lambda x}{\left (x^{2}+a \right )^{2}}+\left (-\frac {x b}{x^{2}+a}-\frac {c}{x \left (x^{2}+a \right )}\right ) \left (-\frac {x^{2} \lambda }{x^{2}+a}-\frac {a \lambda }{x^{2}+a}\right )\right ) u^{\prime }\left (x \right )-\frac {\left (-\frac {x^{2} \lambda }{x^{2}+a}-\frac {a \lambda }{x^{2}+a}\right )^{2} s u \left (x \right )}{x^{2}+a} = 0 \]

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 {a -c}{a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )+c_2 \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right ) \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = \frac {c_1 \,x^{\frac {a -c}{a}} \left (a -c \right ) \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{a x}+\frac {c_1 \,x^{\frac {a -c}{a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (\left (-b +2\right ) a +c \right ) x \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{a \left (x^{2}+a \right )}-\frac {4 c_1 \,x^{\frac {a -c}{a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ) \left (-\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ) \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {9}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {9}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}+1\right ], -\frac {x^{2}}{a}\right ) x}{3 a -c}+\frac {c_2 \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (\left (-b +2\right ) a +c \right ) x \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{a \left (x^{2}+a \right )}+\frac {c_2 \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c \right ) \left (\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c \right ) \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}+1, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}+1\right ], \left [\frac {a +c}{2 a}+1\right ], -\frac {x^{2}}{a}\right ) x}{4 a^{2} \left (a +c \right )} \end{equation}

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u \left (-\frac {x^{2} \lambda }{x^{2}+a}-\frac {a \lambda }{x^{2}+a}\right )} \\ y &= -\frac {\frac {c_1 \,x^{\frac {a -c}{a}} \left (a -c \right ) \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{a x}+\frac {c_1 \,x^{\frac {a -c}{a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (\left (-b +2\right ) a +c \right ) x \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{a \left (x^{2}+a \right )}-\frac {4 c_1 \,x^{\frac {a -c}{a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ) \left (-\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ) \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {9}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {9}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}+1\right ], -\frac {x^{2}}{a}\right ) x}{3 a -c}+\frac {c_2 \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (\left (-b +2\right ) a +c \right ) x \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{a \left (x^{2}+a \right )}+\frac {c_2 \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c \right ) \left (\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c \right ) \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}+1, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}+1\right ], \left [\frac {a +c}{2 a}+1\right ], -\frac {x^{2}}{a}\right ) x}{4 a^{2} \left (a +c \right )}}{\left (-\frac {x^{2} \lambda }{x^{2}+a}-\frac {a \lambda }{x^{2}+a}\right ) \left (c_1 \,x^{\frac {a -c}{a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )+c_2 \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )\right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {\frac {x^{\frac {a -c}{a}} \left (a -c \right ) \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{a x}+\frac {x^{\frac {a -c}{a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (\left (-b +2\right ) a +c \right ) x \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{a \left (x^{2}+a \right )}-\frac {4 x^{\frac {a -c}{a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ) \left (-\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ) \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {9}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {9}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}+1\right ], -\frac {x^{2}}{a}\right ) x}{3 a -c}+\frac {c_3 \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (\left (-b +2\right ) a +c \right ) x \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{a \left (x^{2}+a \right )}+\frac {c_3 \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \left (\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c \right ) \left (\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c \right ) \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}+1, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}+1\right ], \left [\frac {a +c}{2 a}+1\right ], -\frac {x^{2}}{a}\right ) x}{4 a^{2} \left (a +c \right )}}{\left (-\frac {x^{2} \lambda }{x^{2}+a}-\frac {a \lambda }{x^{2}+a}\right ) \left (x^{\frac {a -c}{a}} \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )+c_3 \left (x^{2}+a \right )^{\frac {\left (-b +2\right ) a +c}{2 a}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )\right )} \]

Simplifying the above gives

\begin{align*} y &= \frac {c_3 \left (a -\frac {c}{3}\right ) \left (a \,x^{\frac {a +c}{a}}+x^{\frac {3 a +c}{a}}\right ) \left (\left (-\lambda s +b -2\right ) a^{2}+c \left (b -3\right ) a -c^{2}\right ) \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -7 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +7\right ) a +2 c}{4 a}\right ], \left [\frac {3 a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )+a \left (a +c \right ) \left (-\left (\left (b -2\right ) a -c \right ) x^{\frac {a +c}{a}} c_3 \left (a -\frac {c}{3}\right ) \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )+a \left (\frac {2 \left (-\frac {\lambda s}{2}+b -3\right ) \left (x^{2}+a \right ) x^{2} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {9}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {9}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {5 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{3}+\left (a -\frac {c}{3}\right ) \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right ) \left (a -c +\left (-b +3\right ) x^{2}\right )\right )\right )}{a^{2} \lambda \left (x \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_3 \,x^{\frac {c}{a}}\right ) \left (x^{2}+a \right ) \left (a -\frac {c}{3}\right ) \left (a +c \right )} \\ \end{align*}

Summary of solutions found

\begin{align*} y &= \frac {c_3 \left (a -\frac {c}{3}\right ) \left (a \,x^{\frac {a +c}{a}}+x^{\frac {3 a +c}{a}}\right ) \left (\left (-\lambda s +b -2\right ) a^{2}+c \left (b -3\right ) a -c^{2}\right ) \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -7 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +7\right ) a +2 c}{4 a}\right ], \left [\frac {3 a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )+a \left (a +c \right ) \left (-\left (\left (b -2\right ) a -c \right ) x^{\frac {a +c}{a}} c_3 \left (a -\frac {c}{3}\right ) \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right )+a \left (\frac {2 \left (-\frac {\lambda s}{2}+b -3\right ) \left (x^{2}+a \right ) x^{2} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {9}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {9}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {5 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{3}+\left (a -\frac {c}{3}\right ) \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right ) \left (a -c +\left (-b +3\right ) x^{2}\right )\right )\right )}{a^{2} \lambda \left (x \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +a b -3 a -2 c}{4 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}\, a +\left (-b +3\right ) a +2 c}{4 a}\right ], \left [\frac {a +c}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_3 \,x^{\frac {c}{a}}\right ) \left (x^{2}+a \right ) \left (a -\frac {c}{3}\right ) \left (a +c \right )} \\ \end{align*}

2.2.64.2 ✓ Maple. Time used: 0.057 (sec). Leaf size: 613

ode:=x*(x^2+a)*(diff(y(x),x)+lambda*y(x)^2)+(b*x^2+c)*y(x)+s*x = 0; 
dsolve(ode,y(x), singsol=all);
 

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

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/(x^2+a) 
*diff(y(x),x)-lambda*s/(x^2+a)*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 
         -> hypergeometric 
            -> heuristic approach 
            <- heuristic approach successful 
         <- hypergeometric successful 
      <- special function solution successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (x^{2}+a \right ) \left (\frac {d}{d x}y \left (x \right )+\lambda y \left (x \right )^{2}\right )+\left (b \,x^{2}+c \right ) y \left (x \right )+s x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\frac {y \left (x \right )^{2} \lambda \,x^{3}+y \left (x \right )^{2} a \lambda x +y \left (x \right ) b \,x^{2}+y \left (x \right ) c +s x}{x \left (x^{2}+a \right )} \end {array} \]

2.2.64.3 ✓ Mathematica. Time used: 1.109 (sec). Leaf size: 862

ode=x*(x^2+a)*(D[y[x],x]+\[Lambda]*y[x]^2)+(b*x^2+c)*y[x]+s*x==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

2.2.64.4 ✗ Sympy

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

Timed Out