Internal problem ID [10341]
Internal file name [OUTPUT/9289_Monday_June_06_2022_01_49_06_PM_3413175/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: 12.
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]
\[ \boxed {\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )=-a_{0} x -b_{0}} \]
In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= -\frac {y^{2} a_{2} \lambda x +y^{2} b_{2} \lambda +a_{0} x +b_{0}}{a_{2} x +b_{2}} \end {align*}
This is a Riccati ODE. Comparing the ODE to solve \[ y' = -\frac {y^{2} a_{2} \lambda x}{a_{2} x +b_{2}}-\frac {y^{2} b_{2} \lambda }{a_{2} x +b_{2}}-\frac {a_{0} x}{a_{2} x +b_{2}}-\frac {b_{0}}{a_{2} x +b_{2}} \] With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \] Shows that \(f_0(x)=-\frac {a_{0} x +b_{0}}{a_{2} x +b_{2}}\), \(f_1(x)=0\) and \(f_2(x)=-\frac {a_{2} \lambda x +b_{2} \lambda }{a_{2} x +b_{2}}\). Let \begin {align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-\frac {\left (a_{2} \lambda x +b_{2} \lambda \right ) u}{a_{2} x +b_{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 {a_{2} \lambda }{a_{2} x +b_{2}}+\frac {\left (a_{2} \lambda x +b_{2} \lambda \right ) a_{2}}{\left (a_{2} x +b_{2} \right )^{2}}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=-\frac {\left (a_{2} \lambda x +b_{2} \lambda \right )^{2} \left (a_{0} x +b_{0} \right )}{\left (a_{2} x +b_{2} \right )^{3}} \end {align*}
Substituting the above terms back in equation (2) gives \begin {align*} -\frac {\left (a_{2} \lambda x +b_{2} \lambda \right ) u^{\prime \prime }\left (x \right )}{a_{2} x +b_{2}}-\left (-\frac {a_{2} \lambda }{a_{2} x +b_{2}}+\frac {\left (a_{2} \lambda x +b_{2} \lambda \right ) a_{2}}{\left (a_{2} x +b_{2} \right )^{2}}\right ) u^{\prime }\left (x \right )-\frac {\left (a_{2} \lambda x +b_{2} \lambda \right )^{2} \left (a_{0} x +b_{0} \right ) u \left (x \right )}{\left (a_{2} x +b_{2} \right )^{3}} &=0 \end {align*}
Solving the above ODE (this ode solved using Maple, not this program), gives
\[ u \left (x \right ) = c_{1} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+c_{2} \operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right ) \] The above shows that \[ u^{\prime }\left (x \right ) = \frac {\frac {c_{1} \left (2 a_{2}^{\frac {3}{2}} a_{0}^{\frac {3}{2}}+i \left (a_{0} b_{2} -a_{2} b_{0} \right ) a_{0} \sqrt {\lambda }\right ) \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}+1, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )}{2}-a_{2}^{\frac {3}{2}} a_{0}^{\frac {3}{2}} \operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}+1, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right ) c_{2} +i \left (\left (a_{2} x +\frac {b_{2}}{2}\right ) a_{0} +\frac {a_{2} b_{0}}{2}\right ) a_{0} \sqrt {\lambda }\, \left (c_{1} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+c_{2} \operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )\right )}{\sqrt {a_{2}}\, a_{0}^{\frac {3}{2}} \left (a_{2} x +b_{2} \right )} \] Using the above in (1) gives the solution \[ y = \frac {\frac {c_{1} \left (2 a_{2}^{\frac {3}{2}} a_{0}^{\frac {3}{2}}+i \left (a_{0} b_{2} -a_{2} b_{0} \right ) a_{0} \sqrt {\lambda }\right ) \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}+1, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )}{2}-a_{2}^{\frac {3}{2}} a_{0}^{\frac {3}{2}} \operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}+1, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right ) c_{2} +i \left (\left (a_{2} x +\frac {b_{2}}{2}\right ) a_{0} +\frac {a_{2} b_{0}}{2}\right ) a_{0} \sqrt {\lambda }\, \left (c_{1} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+c_{2} \operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )\right )}{\sqrt {a_{2}}\, a_{0}^{\frac {3}{2}} \left (a_{2} \lambda x +b_{2} \lambda \right ) \left (c_{1} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+c_{2} \operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\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 {\frac {\left (2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )\right ) c_{3} \operatorname {WhittakerM}\left (\frac {2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )}{2}-a_{2}^{\frac {3}{2}} \operatorname {WhittakerW}\left (\frac {2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right ) \sqrt {a_{0}}+i \left (c_{3} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+\operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )\right ) \left (\left (a_{0} x +\frac {b_{0}}{2}\right ) a_{2} +\frac {a_{0} b_{2}}{2}\right ) \sqrt {\lambda }}{\sqrt {a_{0}}\, \sqrt {a_{2}}\, \left (a_{2} x +b_{2} \right ) \lambda \left (c_{3} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+\operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )\right )} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\frac {\left (2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )\right ) c_{3} \operatorname {WhittakerM}\left (\frac {2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )}{2}-a_{2}^{\frac {3}{2}} \operatorname {WhittakerW}\left (\frac {2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right ) \sqrt {a_{0}}+i \left (c_{3} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+\operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )\right ) \left (\left (a_{0} x +\frac {b_{0}}{2}\right ) a_{2} +\frac {a_{0} b_{2}}{2}\right ) \sqrt {\lambda }}{\sqrt {a_{0}}\, \sqrt {a_{2}}\, \left (a_{2} x +b_{2} \right ) \lambda \left (c_{3} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+\operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )\right )} \\ \end{align*}
Verification of solutions
\[ y = \frac {\frac {\left (2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )\right ) c_{3} \operatorname {WhittakerM}\left (\frac {2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )}{2}-a_{2}^{\frac {3}{2}} \operatorname {WhittakerW}\left (\frac {2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}+i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right ) \sqrt {a_{0}}+i \left (c_{3} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+\operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )\right ) \left (\left (a_{0} x +\frac {b_{0}}{2}\right ) a_{2} +\frac {a_{0} b_{2}}{2}\right ) \sqrt {\lambda }}{\sqrt {a_{0}}\, \sqrt {a_{2}}\, \left (a_{2} x +b_{2} \right ) \lambda \left (c_{3} \operatorname {WhittakerM}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )+\operatorname {WhittakerW}\left (\frac {i \sqrt {\lambda }\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{2}^{\frac {3}{2}} \sqrt {a_{0}}}, \frac {1}{2}, \frac {2 i \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}{a_{2}^{\frac {3}{2}}}\right )\right )} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )=-a_{0} x -b_{0} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y^{2} a_{2} \lambda x +y^{2} b_{2} \lambda +a_{0} x +b_{0}}{a_{2} x +b_{2}} \end {array} \]
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: <- Abel AIR successful: ODE belongs to the 1F1 2-parameter class`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 461
dsolve((a__2*x+b__2)*(diff(y(x),x)+lambda*y(x)^2)+a__0*x+b__0=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (\frac {c_{1} \lambda \left (a_{0} b_{2} -a_{2} b_{0} \right ) \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )}{2}+\sqrt {-a_{2} \lambda a_{0}}\, a_{2} \left (c_{1} \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )+\lambda \operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )+\operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right ) \lambda \right )\right ) a_{0}}{\left (\frac {c_{1} \sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right ) \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )}{2}+a_{0} a_{2}^{2} \left (\operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right ) \lambda +c_{1} \operatorname {KummerU}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, \left (a_{0} b_{2} -a_{2} b_{0} \right )}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )-\lambda \operatorname {KummerM}\left (-\frac {\sqrt {-a_{2} \lambda a_{0}}\, a_{0} b_{2} -\sqrt {-a_{2} \lambda a_{0}}\, a_{2} b_{0} -2 a_{0} a_{2}^{2}}{2 a_{0} a_{2}^{2}}, 1, \frac {2 \left (a_{2} x +b_{2} \right ) \sqrt {-a_{2} \lambda a_{0}}}{a_{2}^{2}}\right )\right )\right ) \lambda } \]
✓ Solution by Mathematica
Time used: 1.895 (sec). Leaf size: 690
DSolve[(a2*x+b2)*(y'[x]+\[Lambda]*y[x]^2)+a0*x+b0==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2}) \operatorname {HypergeometricU}\left (\frac {i \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2})}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}+1,1,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )-i \sqrt {\text {a0}} \text {a2}^{3/2} \left (c_1 \operatorname {HypergeometricU}\left (\frac {i (\text {a2} \text {b0}-\text {a0} \text {b2}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}},0,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )+2 \operatorname {LaguerreL}\left (\frac {i (\text {a0} \text {b2}-\text {a2} \text {b0}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}-1,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )+L_{\frac {i (\text {a0} \text {b2}-\text {a2} \text {b0}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}}^{-1}\left (\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )\right )}{\text {a2}^2 \sqrt {\lambda } \left (c_1 \operatorname {HypergeometricU}\left (\frac {i (\text {a2} \text {b0}-\text {a0} \text {b2}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}},0,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )+L_{\frac {i (\text {a0} \text {b2}-\text {a2} \text {b0}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}}^{-1}\left (\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )\right )} \\ y(x)\to \frac {(\text {a2} \text {b0}-\text {a0} \text {b2}) \operatorname {HypergeometricU}\left (\frac {i \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2})}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}+1,1,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )}{\text {a2}^2 \operatorname {HypergeometricU}\left (\frac {i (\text {a2} \text {b0}-\text {a0} \text {b2}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}},0,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )}-\frac {i \sqrt {\text {a0}}}{\sqrt {\text {a2}} \sqrt {\lambda }} \\ y(x)\to \frac {(\text {a2} \text {b0}-\text {a0} \text {b2}) \operatorname {HypergeometricU}\left (\frac {i \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2})}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}+1,1,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )}{\text {a2}^2 \operatorname {HypergeometricU}\left (\frac {i (\text {a2} \text {b0}-\text {a0} \text {b2}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}},0,\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )}-\frac {i \sqrt {\text {a0}}}{\sqrt {\text {a2}} \sqrt {\lambda }} \\ \end{align*}