Internal problem ID [10346]
Internal file name [OUTPUT/9294_Monday_June_06_2022_01_49_17_PM_87108479/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: 17.
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]
\[ \boxed {\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )=-a_{0}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {c_{2} \lambda \,x^{2} y^{2}+y^{2} b_{2} \lambda x +y^{2} a_{2} \lambda +a_{0}}{c_{2} x^{2}+b_{2} x +a_{2}} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {y^{2} c_{2} \lambda \,x^{2}}{c_{2} x^{2}+b_{2} x +a_{2}}-\frac {y^{2} b_{2} \lambda x}{c_{2} x^{2}+b_{2} x +a_{2}}-\frac {y^{2} a_{2} \lambda }{c_{2} x^{2}+b_{2} x +a_{2}}-\frac {a_{0}}{c_{2} x^{2}+b_{2} x +a_{2}} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {a_{0}}{c_{2} x^{2}+b_{2} x +a_{2}}\), \(f_1(x)=0\) and \(f_2(x)=-\frac {c_{2} \lambda \,x^{2}+b_{2} \lambda x +a_{2} \lambda }{c_{2} x^{2}+b_{2} x +a_{2}}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {\left (c_{2} \lambda \,x^{2}+b_{2} \lambda x +a_{2} \lambda \right ) u}{c_{2} x^{2}+b_{2} x +a_{2}}} \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 c_{2} \lambda x +b_{2} \lambda }{c_{2} x^{2}+b_{2} x +a_{2}}+\frac {\left (c_{2} \lambda \,x^{2}+b_{2} \lambda x +a_{2} \lambda \right ) \left (2 c_{2} x +b_{2} \right )}{\left (c_{2} x^{2}+b_{2} x +a_{2} \right )^{2}}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=-\frac {\left (c_{2} \lambda \,x^{2}+b_{2} \lambda x +a_{2} \lambda \right )^{2} a_{0}}{\left (c_{2} x^{2}+b_{2} x +a_{2} \right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\frac {\left (c_{2} \lambda \,x^{2}+b_{2} \lambda x +a_{2} \lambda \right ) u^{\prime \prime }\left (x \right )}{c_{2} x^{2}+b_{2} x +a_{2}}-\left (-\frac {2 c_{2} \lambda x +b_{2} \lambda }{c_{2} x^{2}+b_{2} x +a_{2}}+\frac {\left (c_{2} \lambda \,x^{2}+b_{2} \lambda x +a_{2} \lambda \right ) \left (2 c_{2} x +b_{2} \right )}{\left (c_{2} x^{2}+b_{2} x +a_{2} \right )^{2}}\right ) u^{\prime }\left (x \right )-\frac {\left (c_{2} \lambda \,x^{2}+b_{2} \lambda x +a_{2} \lambda \right )^{2} a_{0} u \left (x \right )}{\left (c_{2} x^{2}+b_{2} x +a_{2} \right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = -2 \left (c_{4} \left (2 c_{2} x +b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\right )^{-\frac {\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}} \operatorname {hypergeom}\left (\left [\frac {\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}, \frac {3 \sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}\right ], \left [\frac {\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{\sqrt {c_{2}}}\right ], \frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 c_{2} x +b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}\right )+c_{3} \left (2 c_{2} x +b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\right )^{\frac {-\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}} \operatorname {hypergeom}\left (\left [-\frac {-3 \sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}, -\frac {-\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}\right ], \left [\frac {\sqrt {c_{2}}-\sqrt {-4 a_{0} \lambda +c_{2}}}{\sqrt {c_{2}}}\right ], \frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 c_{2} x +b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}\right )\right ) \left (c_{2} x -\frac {\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2}+\frac {b_{2}}{2}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = -\frac {4 \left (c_{3} \left (\left (\left (c_{2} x +\frac {b_{2}}{2}\right ) \sqrt {-4 a_{0} \lambda +c_{2}}+c_{2}^{\frac {3}{2}} x +\frac {\sqrt {c_{2}}\, b_{2}}{2}\right ) \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}+\left (\frac {b_{2}^{2}}{2}+c_{2} b_{2} x +c_{2} \left (c_{2} x^{2}-a_{2} \right )\right ) \sqrt {-4 a_{0} \lambda +c_{2}}+\left (b_{2} x -a_{2} \right ) c_{2}^{\frac {3}{2}}+c_{2}^{\frac {5}{2}} x^{2}+\frac {b_{2}^{2} \sqrt {c_{2}}}{2}\right ) \left (2 c_{2} x +b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\right )^{\frac {-\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}} \operatorname {hypergeom}\left (\left [-\frac {-\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}, -\frac {-\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}\right ], \left [\frac {\sqrt {c_{2}}-\sqrt {-4 a_{0} \lambda +c_{2}}}{\sqrt {c_{2}}}\right ], \frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 c_{2} x +b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}\right )+\operatorname {hypergeom}\left (\left [\frac {\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}, \frac {\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}\right ], \left [\frac {\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{\sqrt {c_{2}}}\right ], \frac {2 \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}{2 c_{2} x +b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}}\right ) \left (2 c_{2} x +b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\right )^{-\frac {\sqrt {c_{2}}+\sqrt {-4 a_{0} \lambda +c_{2}}}{2 \sqrt {c_{2}}}} \left (\left (\left (-c_{2} x -\frac {b_{2}}{2}\right ) \sqrt {-4 a_{0} \lambda +c_{2}}+c_{2}^{\frac {3}{2}} x +\frac {\sqrt {c_{2}}\, b_{2}}{2}\right ) \sqrt {-4 a_{2} c_{2} +b_{2}^{2}}+\left (-c_{2}^{2} x^{2}-c_{2} b_{2} x +a_{2} c_{2} -\frac {1}{2} b_{2}^{2}\right ) \sqrt {-4 a_{0} \lambda +c_{2}}+\left (b_{2} x -a_{2} \right ) c_{2}^{\frac {3}{2}}+c_{2}^{\frac {5}{2}} x^{2}+\frac {b_{2}^{2} \sqrt {c_{2}}}{2}\right ) c_{4} \right ) \sqrt {c_{2}}}{\left (2 c_{2} x +b_{2} +\sqrt {-4 a_{2} c_{2} +b_{2}^{2}}\right )^{2}} \] Using the above in (1) gives the solution \[ \text {Expression too large to display} \] Dividing both numerator and denominator by \(c_{1}\) gives, after renaming the constant \(\frac {c_{2}}{c_{1}}=c_{3}\) the following solution
\[ y = -\frac {4 \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1+\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right ) c_{4} \left (-1+\sqrt {-4 a_{0} \lambda +1}\right ) \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}}-\left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1-\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right ) c_{3} \left (1+\sqrt {-4 a_{0} \lambda +1}\right )\right ) \left (\left (x +\frac {b_{2}}{2}\right ) \sqrt {b_{2}^{2}-4 a_{2}}+x^{2}+b_{2} x +\frac {b_{2}^{2}}{2}-a_{2} \right )}{\left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{2} \left (-\sqrt {b_{2}^{2}-4 a_{2}}+2 x +b_{2} \right ) \lambda \left (c_{4} \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {3}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1+\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right )+c_{3} \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {3}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1-\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {4 \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1+\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right ) c_{4} \left (-1+\sqrt {-4 a_{0} \lambda +1}\right ) \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}}-\left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1-\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right ) c_{3} \left (1+\sqrt {-4 a_{0} \lambda +1}\right )\right ) \left (\left (x +\frac {b_{2}}{2}\right ) \sqrt {b_{2}^{2}-4 a_{2}}+x^{2}+b_{2} x +\frac {b_{2}^{2}}{2}-a_{2} \right )}{\left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{2} \left (-\sqrt {b_{2}^{2}-4 a_{2}}+2 x +b_{2} \right ) \lambda \left (c_{4} \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {3}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1+\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right )+c_{3} \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {3}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1-\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = -\frac {4 \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1+\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right ) c_{4} \left (-1+\sqrt {-4 a_{0} \lambda +1}\right ) \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}}-\left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1-\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right ) c_{3} \left (1+\sqrt {-4 a_{0} \lambda +1}\right )\right ) \left (\left (x +\frac {b_{2}}{2}\right ) \sqrt {b_{2}^{2}-4 a_{2}}+x^{2}+b_{2} x +\frac {b_{2}^{2}}{2}-a_{2} \right )}{\left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{2} \left (-\sqrt {b_{2}^{2}-4 a_{2}}+2 x +b_{2} \right ) \lambda \left (c_{4} \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {3}{2}+\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1+\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right )+c_{3} \left (2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}\right )^{\frac {\sqrt {-4 a_{0} \lambda +1}}{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}, \frac {3}{2}-\frac {\sqrt {-4 a_{0} \lambda +1}}{2}\right ], \left [1-\sqrt {-4 a_{0} \lambda +1}\right ], \frac {2 \sqrt {b_{2}^{2}-4 a_{2}}}{2 x +b_{2} +\sqrt {b_{2}^{2}-4 a_{2}}}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )=-a_{0} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y^{2} c_{2} \lambda \,x^{2}+y^{2} b_{2} \lambda x +y^{2} a_{2} \lambda +a_{0}}{c_{2} x^{2}+b_{2} x +a_{2}} \end {array} \]
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__0*lambda*y(x)/(c__2*x^2+b__2*x+a__2), 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 -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 2F1 ODE <- hypergeometric successful <- special function solution successful <- Riccati to 2nd Order successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 1127
dsolve((c__2*x^2+b__2*x+a__2)*(diff(y(x),x)+lambda*y(x)^2)+a__0=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 5.964 (sec). Leaf size: 1046
DSolve[(c2*x^2+b2*x+a2)*(y'[x]+\[Lambda]*y[x]^2)+a0==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(\text {b2}+2 \text {c2} x) \left (8 \text {c2} \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) G_{2,2}^{2,0}\left (-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}| \begin {array}{c} \frac {1}{4}-\frac {\sqrt {\text {c2}-4 \text {a0} \lambda }}{4 \sqrt {\text {c2}}},\frac {1}{4} \left (\frac {\sqrt {\text {c2}-4 \text {a0} \lambda }}{\sqrt {\text {c2}}}+1\right ) \\ 0,0 \\ \end {array} \right )+c_1 \left (8 \text {c2} \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )+\left (\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}-3 \text {c2}\right ) \left (\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}+3 \text {c2}\right ) (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {7 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (7-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),3,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )\right )\right )}{2 \lambda \left (\text {b2}^2-4 \text {a2} \text {c2}\right )^2 \left (G_{2,2}^{2,0}\left (-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}| \begin {array}{c} \frac {1}{4} \left (5-\frac {\sqrt {\text {c2}-4 \text {a0} \lambda }}{\sqrt {\text {c2}}}\right ),\frac {1}{4} \left (\frac {\sqrt {\text {c2}-4 \text {a0} \lambda }}{\sqrt {\text {c2}}}+5\right ) \\ 0,1 \\ \end {array} \right )-\frac {4 \text {c2} c_1 (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )}{4 \text {a2} \text {c2}-\text {b2}^2}\right )} \\ y(x)\to \frac {(\text {b2}+2 \text {c2} x) \left (2 \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )-(\text {a0} \lambda +2 \text {c2}) (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {7 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (7-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),3,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )\right )}{2 \lambda \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )} \\ y(x)\to \frac {(\text {b2}+2 \text {c2} x) \left (2 \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )-(\text {a0} \lambda +2 \text {c2}) (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {7 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (7-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),3,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )\right )}{2 \lambda \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )} \\ \end{align*}