Internal problem ID [10379]
Internal file name [OUTPUT/9327_Monday_June_06_2022_01_51_39_PM_55641103/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: 50.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "riccati"
Maple gives the following as the ode type
[_rational, _Riccati]
\[ \boxed {x^{2} y^{\prime }-y^{2} c \,x^{2}-\left (x^{2} a +b x \right ) y=\alpha \,x^{2}+\beta x +\gamma } \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= \frac {y^{2} c \,x^{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 \[ y' = c \,y^{2}+y a +\alpha +\frac {y b}{x}+\frac {\beta }{x}+\frac {\gamma }{x^{2}} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=\frac {\alpha \,x^{2}+\beta x +\gamma }{x^{2}}\), \(f_1(x)=\frac {x^{2} a +b x}{x^{2}}\) 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 (x^{2} a +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 \begin {align*} c u^{\prime \prime }\left (x \right )-\frac {\left (x^{2} a +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 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = {\mathrm e}^{\frac {x a}{2}} x^{\frac {b}{2}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+c_{2} \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {{\mathrm e}^{\frac {x a}{2}} x^{-1+\frac {b}{2}} \left (-c_{1} \left (a b -2 \beta c -\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}-\sqrt {a^{2}-4 \alpha c}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{2} \sqrt {a^{2}-4 \alpha c}+\left (c_{1} \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+c_{2} \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) \left (\left (x a +b \right ) \sqrt {a^{2}-4 \alpha c}+a^{2} x +a b -4 \left (\alpha x +\frac {\beta }{2}\right ) c \right )\right )}{2 \sqrt {a^{2}-4 \alpha c}} \] Using the above in (1) gives the solution \[ y = -\frac {x^{-1+\frac {b}{2}} \left (-c_{1} \left (a b -2 \beta c -\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}-\sqrt {a^{2}-4 \alpha c}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{2} \sqrt {a^{2}-4 \alpha c}+\left (c_{1} \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+c_{2} \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) \left (\left (x a +b \right ) \sqrt {a^{2}-4 \alpha c}+a^{2} x +a b -4 \left (\alpha x +\frac {\beta }{2}\right ) c \right )\right ) x^{-\frac {b}{2}}}{2 \sqrt {a^{2}-4 \alpha c}\, c \left (c_{1} \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+c_{2} \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right )} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {-c_{3} \left (a b -2 \beta c -\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}-\sqrt {a^{2}-4 \alpha c}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) \sqrt {a^{2}-4 \alpha c}+\left (c_{3} \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) \left (\left (x a +b \right ) \sqrt {a^{2}-4 \alpha c}+a^{2} x +a b -4 \left (\alpha x +\frac {\beta }{2}\right ) c \right )}{2 \sqrt {a^{2}-4 \alpha c}\, c x \left (c_{3} \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {-c_{3} \left (a b -2 \beta c -\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha 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 ) \sqrt {a^{2}-4 \alpha c}+\left (c_{3} \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 )+\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 )\right ) \left (\left (x a +b \right ) \sqrt {a^{2}-4 \alpha c}+a^{2} x +a b -4 \left (\alpha x +\frac {\beta }{2}\right ) c \right )}{2 \sqrt {a^{2}-4 \alpha c}\, c x \left (c_{3} \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 )+\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 )\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {-c_{3} \left (a b -2 \beta c -\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}-\sqrt {a^{2}-4 \alpha c}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) \sqrt {a^{2}-4 \alpha c}+\left (c_{3} \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) \left (\left (x a +b \right ) \sqrt {a^{2}-4 \alpha c}+a^{2} x +a b -4 \left (\alpha x +\frac {\beta }{2}\right ) c \right )}{2 \sqrt {a^{2}-4 \alpha c}\, c x \left (c_{3} \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )+\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }-y^{2} c \,x^{2}-\left (x^{2} a +b x \right ) y=\alpha \,x^{2}+\beta x +\gamma \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2} c \,x^{2}+\left (x^{2} a +b x \right ) y+\alpha \,x^{2}+\beta x +\gamma }{x^{2}} \end {array} \]
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) = (a*x+b)*(diff(y(x), x))/x-c*(alpha*x^2+beta*x+gamma)*y(x)/x^2, 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 -> 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`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 443
dsolve(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,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (\sqrt {b^{2}-4 c \gamma +2 b +1}\, \sqrt {a^{2}-4 \alpha c}-a b +2 \beta c +\sqrt {a^{2}-4 \alpha c}\right ) \operatorname {WhittakerM}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )-2 \operatorname {WhittakerW}\left (-\frac {a b -2 \beta c -2 \sqrt {a^{2}-4 \alpha c}}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{1} \sqrt {a^{2}-4 \alpha c}+\left (\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) \left (\left (a x +b \right ) \sqrt {a^{2}-4 \alpha c}+a^{2} x +a b -4 \left (\alpha x +\frac {\beta }{2}\right ) c \right )}{2 \sqrt {a^{2}-4 \alpha c}\, \left (\operatorname {WhittakerW}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {a b -2 \beta c}{2 \sqrt {a^{2}-4 \alpha c}}, \frac {\sqrt {b^{2}-4 c \gamma +2 b +1}}{2}, \sqrt {a^{2}-4 \alpha c}\, x \right )\right ) c x} \]
✓ Solution by Mathematica
Time used: 1.712 (sec). Leaf size: 1584
DSolve[x^2*y'[x]==c*x^2*y[x]^2+(a*x^2+b*x)*y[x]+\[Alpha]*x^2+\[Beta]*x+\[Gamma],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\left (b+a x-x \sqrt {a^2-4 c \alpha }+\sqrt {b^2+2 b-4 c \gamma +1}+1\right ) c_1 \operatorname {HypergeometricU}\left (\frac {a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }},\sqrt {b^2+2 b-4 c \gamma +1}+1,x \sqrt {a^2-4 c \alpha }\right )-x \left (a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )\right ) c_1 \operatorname {HypergeometricU}\left (\frac {a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+3\right )}{2 \sqrt {a^2-4 c \alpha }},\sqrt {b^2+2 b-4 c \gamma +1}+2,x \sqrt {a^2-4 c \alpha }\right )+b L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )+a x L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )-x \sqrt {a^2-4 c \alpha } L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )+\sqrt {b^2+2 b-4 c \gamma +1} L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )+L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )-2 x \sqrt {a^2-4 c \alpha } L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+3\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}+1}\left (x \sqrt {a^2-4 c \alpha }\right )}{2 c x \left (c_1 \operatorname {HypergeometricU}\left (\frac {a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }},\sqrt {b^2+2 b-4 c \gamma +1}+1,x \sqrt {a^2-4 c \alpha }\right )+L_{\frac {-a b+2 c \beta -\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }}}^{\sqrt {b^2+2 b-4 c \gamma +1}}\left (x \sqrt {a^2-4 c \alpha }\right )\right )} \\ y(x)\to \frac {\frac {\left (\sqrt {a^2-4 \alpha c} \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )+a b-2 \beta c\right ) \operatorname {HypergeometricU}\left (\frac {a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+3\right )}{2 \sqrt {a^2-4 c \alpha }},\sqrt {b^2+2 b-4 c \gamma +1}+2,x \sqrt {a^2-4 c \alpha }\right )}{\operatorname {HypergeometricU}\left (\frac {a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }},\sqrt {b^2+2 b-4 c \gamma +1}+1,x \sqrt {a^2-4 c \alpha }\right )}-\frac {-x \sqrt {a^2-4 \alpha c}+a x+\sqrt {b^2+2 b-4 c \gamma +1}+b+1}{x}}{2 c} \\ y(x)\to \frac {\frac {\left (\sqrt {a^2-4 \alpha c} \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )+a b-2 \beta c\right ) \operatorname {HypergeometricU}\left (\frac {a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+3\right )}{2 \sqrt {a^2-4 c \alpha }},\sqrt {b^2+2 b-4 c \gamma +1}+2,x \sqrt {a^2-4 c \alpha }\right )}{\operatorname {HypergeometricU}\left (\frac {a b-2 c \beta +\sqrt {a^2-4 c \alpha } \left (\sqrt {b^2+2 b-4 c \gamma +1}+1\right )}{2 \sqrt {a^2-4 c \alpha }},\sqrt {b^2+2 b-4 c \gamma +1}+1,x \sqrt {a^2-4 c \alpha }\right )}-\frac {-x \sqrt {a^2-4 \alpha c}+a x+\sqrt {b^2+2 b-4 c \gamma +1}+b+1}{x}}{2 c} \\ \end{align*}