problem 67

Internal problem ID [10396]
Internal ﬁle name [OUTPUT/9344_Monday_June_06_2022_02_14_16_PM_45730703/index.tex]

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.
ODE order: 1.
ODE degree: 1.

\[ \boxed {x \left (x^{2}+a \right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (b \,x^{2}+c \right ) y=-s x} \]

2.67.1 Solving as riccati ode

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 +y c +s x}{x \left (x^{2}+a \right )} \end {align*}

This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {x^{2} y^{2} \lambda }{x^{2}+a}-\frac {y^{2} a \lambda }{x^{2}+a}-\frac {x b y}{x^{2}+a}-\frac {y c}{x \left (x^{2}+a \right )}-\frac {s}{x^{2}+a} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {s}{x^{2}+a}\), \(f_1(x)=-\frac {b \,x^{2}+c}{x \left (x^{2}+a \right )}\) and \(f_2(x)=-\frac {\lambda \,x^{3}+a \lambda x}{x \left (x^{2}+a \right )}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {\left (\lambda \,x^{3}+a \lambda x \right ) u}{x \left (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 {3 \lambda \,x^{2}+a \lambda }{x \left (x^{2}+a \right )}+\frac {\lambda \,x^{3}+a \lambda x}{x^{2} \left (x^{2}+a \right )}+\frac {2 \lambda \,x^{3}+2 a \lambda x}{\left (x^{2}+a \right )^{2}}\\ f_1 f_2 &=\frac {\left (b \,x^{2}+c \right ) \left (\lambda \,x^{3}+a \lambda x \right )}{x^{2} \left (x^{2}+a \right )^{2}}\\ f_2^2 f_0 &=-\frac {\left (\lambda \,x^{3}+a \lambda x \right )^{2} s}{x^{2} \left (x^{2}+a \right )^{3}} \end {align*}

Substituting the above terms back in equation (2) gives \begin {align*} -\frac {\left (\lambda \,x^{3}+a \lambda x \right ) u^{\prime \prime }\left (x \right )}{x \left (x^{2}+a \right )}-\left (-\frac {3 \lambda \,x^{2}+a \lambda }{x \left (x^{2}+a \right )}+\frac {\lambda \,x^{3}+a \lambda x}{x^{2} \left (x^{2}+a \right )}+\frac {2 \lambda \,x^{3}+2 a \lambda x}{\left (x^{2}+a \right )^{2}}+\frac {\left (b \,x^{2}+c \right ) \left (\lambda \,x^{3}+a \lambda x \right )}{x^{2} \left (x^{2}+a \right )^{2}}\right ) u^{\prime }\left (x \right )-\frac {\left (\lambda \,x^{3}+a \lambda x \right )^{2} s u \left (x \right )}{x^{2} \left (x^{2}+a \right )^{3}} &=0 \end {align*}

\[ u \left (x \right ) = \left (x^{2}+a \right )^{\frac {\left (2-b \right ) a +c}{2 a}} \left (x^{\frac {a -c}{a}} \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_{1} +\operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {3}{4}+\frac {c}{2 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {3}{4}+\frac {c}{2 a}\right ], \left [\frac {1}{2}+\frac {c}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_{2} \right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {3 \left (\left (a -\frac {c}{3}\right ) \left (\left (-\lambda s +b -2\right ) a^{2}+c \left (-3+b \right ) a -c^{2}\right ) \left (x^{2}+a \right ) x^{2} c_{2} \operatorname {hypergeom}\left (\left [\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {7}{4}+\frac {c}{2 a}, -\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {7}{4}+\frac {c}{2 a}\right ], \left [\frac {3}{2}+\frac {c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\left (a +c \right ) a \left (-\left (\left (-2+b \right ) a -c \right ) \left (a -\frac {c}{3}\right ) x^{2} c_{2} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {3}{4}+\frac {c}{2 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {3}{4}+\frac {c}{2 a}\right ], \left [\frac {1}{2}+\frac {c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\left (\frac {2 \left (-\frac {\lambda s}{2}+b -3\right ) \left (a \,x^{\frac {3 a -c}{a}}+x^{\frac {5 a -c}{a}}\right ) \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {9}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {9}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {5 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{3}+\left (a -\frac {c}{3}\right ) \operatorname {hypergeom}\left (\left [-\frac {b}{4}+\frac {5}{4}+\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}, -\frac {b}{4}+\frac {5}{4}-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}\right ], \left [\frac {3 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right ) \left (\left (-b +3\right ) x^{\frac {3 a -c}{a}}+x^{\frac {a -c}{a}} \left (a -c \right )\right )\right ) c_{1} a \right )\right ) \left (x^{2}+a \right )^{\frac {-a b +c}{2 a}}}{a^{2} x \left (a +c \right ) \left (3 a -c \right )} \] Using the above in (1) gives the solution \[ y = \frac {3 \left (\left (a -\frac {c}{3}\right ) \left (\left (-\lambda s +b -2\right ) a^{2}+c \left (-3+b \right ) a -c^{2}\right ) \left (x^{2}+a \right ) x^{2} c_{2} \operatorname {hypergeom}\left (\left [\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {7}{4}+\frac {c}{2 a}, -\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {7}{4}+\frac {c}{2 a}\right ], \left [\frac {3}{2}+\frac {c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\left (a +c \right ) a \left (-\left (\left (-2+b \right ) a -c \right ) \left (a -\frac {c}{3}\right ) x^{2} c_{2} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {3}{4}+\frac {c}{2 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {3}{4}+\frac {c}{2 a}\right ], \left [\frac {1}{2}+\frac {c}{2 a}\right ], -\frac {x^{2}}{a}\right )+\left (\frac {2 \left (-\frac {\lambda s}{2}+b -3\right ) \left (a \,x^{\frac {3 a -c}{a}}+x^{\frac {5 a -c}{a}}\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 {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 (\left (-b +3\right ) x^{\frac {3 a -c}{a}}+x^{\frac {a -c}{a}} \left (a -c \right )\right )\right ) c_{1} a \right )\right ) \left (x^{2}+a \right )^{\frac {-a b +c}{2 a}} \left (x^{2}+a \right ) \left (x^{2}+a \right )^{\frac {a b -2 a -c}{2 a}}}{a^{2} \left (a +c \right ) \left (3 a -c \right ) \left (\lambda \,x^{3}+a \lambda x \right ) \left (x^{\frac {a -c}{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_{1} +\operatorname {hypergeom}\left (\left [-\frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {3}{4}+\frac {c}{2 a}, \frac {\sqrt {b^{2}-4 \lambda s -2 b +1}}{4}-\frac {b}{4}+\frac {3}{4}+\frac {c}{2 a}\right ], \left [\frac {1}{2}+\frac {c}{2 a}\right ], -\frac {x^{2}}{a}\right ) c_{2} \right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution

\[ y = \frac {\left (a -\frac {c}{3}\right ) \left (\left (-\lambda s +b -2\right ) a^{2}+c \left (-3+b \right ) a -c^{2}\right ) \left (a \,x^{\frac {a +c}{a}}+x^{\frac {3 a +c}{a}}\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 )+\left (-\left (\left (-2+b \right ) a -c \right ) \left (a -\frac {c}{3}\right ) x^{\frac {a +c}{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 )+\left (\frac {2 \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 ) \left (-\frac {\lambda s}{2}+b -3\right )}{3}+\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 -\frac {c}{3}\right ) \left (a -c +\left (-b +3\right ) x^{2}\right )\right ) c_{3} a \right ) \left (a +c \right ) a}{\left (a +c \right ) \left (a -\frac {c}{3}\right ) \left (x^{2}+a \right ) \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 ) c_{3} +\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 ) x^{\frac {c}{a}}\right ) \lambda \,a^{2}} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (a -\frac {c}{3}\right ) \left (\left (-\lambda s +b -2\right ) a^{2}+c \left (-3+b \right ) a -c^{2}\right ) \left (a \,x^{\frac {a +c}{a}}+x^{\frac {3 a +c}{a}}\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 )+\left (-\left (\left (-2+b \right ) a -c \right ) \left (a -\frac {c}{3}\right ) x^{\frac {a +c}{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 )+\left (\frac {2 \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 ) \left (-\frac {\lambda s}{2}+b -3\right )}{3}+\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 -\frac {c}{3}\right ) \left (a -c +\left (-b +3\right ) x^{2}\right )\right ) c_{3} a \right ) \left (a +c \right ) a}{\left (a +c \right ) \left (a -\frac {c}{3}\right ) \left (x^{2}+a \right ) \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 ) c_{3} +\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 ) x^{\frac {c}{a}}\right ) \lambda \,a^{2}} \\ \end{align*}

Veriﬁcation of solutions

2.67.2 Maple step by step solution

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

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)*(diff(y(x), x))/(x*(x^2+a))-lambda*s*y(x)/(x^2+a), y(x)` 
      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`

\[ y \left (x \right ) = \frac {\left (a -\frac {c}{3}\right ) \left (\left (-\lambda s +b -2\right ) a^{2}+c \left (b -3\right ) a -c^{2}\right ) \left (a \,x^{\frac {a +c}{a}}+x^{\frac {3 a +c}{a}}\right ) c_{1} \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 )+\left (-\left (a -\frac {c}{3}\right ) \left (\left (b -2\right ) a -c \right ) c_{1} x^{\frac {a +c}{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 )+\left (\frac {2 \left (x^{2}+a \right ) x^{2} \left (-\frac {\lambda s}{2}+b -3\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 {5 a -c}{2 a}\right ], -\frac {x^{2}}{a}\right )}{3}+\left (a -\frac {c}{3}\right ) \left (a -c +\left (-b +3\right ) x^{2}\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 )\right ) a \right ) \left (a +c \right ) a}{\left (a -\frac {c}{3}\right ) \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_{1} x^{\frac {c}{a}}\right ) \left (x^{2}+a \right ) \left (a +c \right ) a^{2} \lambda } \]

\begin{align*} y(x)\to \frac {a^{\frac {1}{2} \left (\frac {c}{a}-3\right )} (a-c) x^{-\frac {c}{a}} \operatorname {Hypergeometric2F1}\left (\frac {b a-\sqrt {b^2-2 b-4 s \lambda +1} a+a-2 c}{4 a},\frac {a \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+1\right )-2 c}{4 a},\frac {3}{2}-\frac {c}{2 a},-\frac {x^2}{a}\right )+\frac {a^{\frac {1}{2} \left (\frac {c}{a}-5\right )} x^{2-\frac {c}{a}} \left (a \left (\sqrt {b^2-2 b-4 \lambda s+1}-b-1\right )+2 c\right ) \left (a \left (\sqrt {b^2-2 b-4 \lambda s+1}+b+1\right )-2 c\right ) \operatorname {Hypergeometric2F1}\left (\frac {a \left (b-\sqrt {b^2-2 b-4 s \lambda +1}+5\right )-2 c}{4 a},\frac {a \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+5\right )-2 c}{4 a},\frac {5}{2}-\frac {c}{2 a},-\frac {x^2}{a}\right )}{4 (3 a-c)}-\frac {c_1 \lambda s x \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{2} \left (\frac {c}{a}+3\right ),-\frac {x^2}{a}\right )}{a+c}}{\lambda a^{\frac {1}{2} \left (\frac {c}{a}-1\right )} x^{1-\frac {c}{a}} \operatorname {Hypergeometric2F1}\left (\frac {b a-\sqrt {b^2-2 b-4 s \lambda +1} a+a-2 c}{4 a},\frac {a \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+1\right )-2 c}{4 a},\frac {3}{2}-\frac {c}{2 a},-\frac {x^2}{a}\right )+c_1 \lambda \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {a+c}{2 a},-\frac {x^2}{a}\right )} \\ y(x)\to -\frac {s x \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{2} \left (\frac {c}{a}+3\right ),-\frac {x^2}{a}\right )}{(a+c) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {a+c}{2 a},-\frac {x^2}{a}\right )} \\ y(x)\to -\frac {s x \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}+3\right ),\frac {1}{2} \left (\frac {c}{a}+3\right ),-\frac {x^2}{a}\right )}{(a+c) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {1}{4} \left (b+\sqrt {b^2-2 b-4 s \lambda +1}-1\right ),\frac {a+c}{2 a},-\frac {x^2}{a}\right )} \\ \end{align*}

2.67 problem 67

2.67.1 Solving as riccati ode

2.67.2 Maple step by step solution