2.2.11 Problem 12

Internal problem ID [13217]
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
Date solved : Wednesday, December 31, 2025 at 12:08:11 PM
CAS classification : [_rational, _Riccati]

2.2.11.1 Solved using first_order_ode_riccati

11.073 (sec)

Entering first order ode riccati solver

\begin{align*} \left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0} x +b_{0}&=0 \\ \end{align*}

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

\[ y' = \textit {the\_rhs} \]

With Riccati ODE standard form

\[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]

Shows that \(f_0(x)=-\frac {a_{0} x}{a_{2} x +b_{2}}-\frac {b_{0}}{a_{2} x +b_{2}}\), \(f_1(x)=0\) and \(f_2(x)=-\frac {a_{2} \lambda x}{a_{2} x +b_{2}}-\frac {b_{2} \lambda }{a_{2} x +b_{2}}\). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u \left (-\frac {a_{2} \lambda x}{a_{2} x +b_{2}}-\frac {b_{2} \lambda }{a_{2} x +b_{2}}\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 {a_{2}^{2} \lambda x}{\left (a_{2} x +b_{2} \right )^{2}}-\frac {a_{2} \lambda }{a_{2} x +b_{2}}+\frac {b_{2} \lambda a_{2}}{\left (a_{2} x +b_{2} \right )^{2}}\\ f_1 f_2 &=0\\ f_2^2 f_0 &=\left (-\frac {a_{2} \lambda x}{a_{2} x +b_{2}}-\frac {b_{2} \lambda }{a_{2} x +b_{2}}\right )^{2} \left (-\frac {a_{0} x}{a_{2} x +b_{2}}-\frac {b_{0}}{a_{2} x +b_{2}}\right ) \end{align*}

Substituting the above terms back in equation (2) gives

\[ \left (-\frac {a_{2} \lambda x}{a_{2} x +b_{2}}-\frac {b_{2} \lambda }{a_{2} x +b_{2}}\right ) u^{\prime \prime }\left (x \right )-\left (\frac {a_{2}^{2} \lambda x}{\left (a_{2} x +b_{2} \right )^{2}}-\frac {a_{2} \lambda }{a_{2} x +b_{2}}+\frac {b_{2} \lambda a_{2}}{\left (a_{2} x +b_{2} \right )^{2}}\right ) u^{\prime }\left (x \right )+\left (-\frac {a_{2} \lambda x}{a_{2} x +b_{2}}-\frac {b_{2} \lambda }{a_{2} x +b_{2}}\right )^{2} \left (-\frac {a_{0} x}{a_{2} x +b_{2}}-\frac {b_{0}}{a_{2} x +b_{2}}\right ) u \left (x \right ) = 0 \]

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 \operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )+c_2 \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right ) \end{equation}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = \frac {2 i c_1 \left (\left (\frac {1}{2}-\frac {a_{0} b_{2} -a_{2} b_{0}}{4 a_{0} \left (a_{2} x +b_{2} \right )}\right ) \operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )-\frac {i \left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{3}/{2}} \sqrt {a_{0}}}+1\right ) a_{2}^{{3}/{2}} \operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )}{2 \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}\right ) \sqrt {a_{0}}\, \sqrt {\lambda }}{\sqrt {a_{2}}}+\frac {2 i c_2 \left (\left (\frac {1}{2}-\frac {a_{0} b_{2} -a_{2} b_{0}}{4 a_{0} \left (a_{2} x +b_{2} \right )}\right ) \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )+\frac {i a_{2}^{{3}/{2}} \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )}{2 \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}\right ) \sqrt {a_{0}}\, \sqrt {\lambda }}{\sqrt {a_{2}}} \end{equation}

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u \left (-\frac {a_{2} \lambda x}{a_{2} x +b_{2}}-\frac {b_{2} \lambda }{a_{2} x +b_{2}}\right )} \\ y &= -\frac {\frac {2 i c_1 \left (\left (\frac {1}{2}-\frac {a_{0} b_{2} -a_{2} b_{0}}{4 a_{0} \left (a_{2} x +b_{2} \right )}\right ) \operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )-\frac {i \left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{3}/{2}} \sqrt {a_{0}}}+1\right ) a_{2}^{{3}/{2}} \operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )}{2 \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}\right ) \sqrt {a_{0}}\, \sqrt {\lambda }}{\sqrt {a_{2}}}+\frac {2 i c_2 \left (\left (\frac {1}{2}-\frac {a_{0} b_{2} -a_{2} b_{0}}{4 a_{0} \left (a_{2} x +b_{2} \right )}\right ) \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )+\frac {i a_{2}^{{3}/{2}} \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )}{2 \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}\right ) \sqrt {a_{0}}\, \sqrt {\lambda }}{\sqrt {a_{2}}}}{\left (-\frac {a_{2} \lambda x}{a_{2} x +b_{2}}-\frac {b_{2} \lambda }{a_{2} x +b_{2}}\right ) \left (c_1 \operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )+c_2 \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )\right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {\frac {2 i \left (\left (\frac {1}{2}-\frac {a_{0} b_{2} -a_{2} b_{0}}{4 a_{0} \left (a_{2} x +b_{2} \right )}\right ) \operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )-\frac {i \left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{3}/{2}} \sqrt {a_{0}}}+1\right ) a_{2}^{{3}/{2}} \operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )}{2 \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}\right ) \sqrt {a_{0}}\, \sqrt {\lambda }}{\sqrt {a_{2}}}+\frac {2 i c_3 \left (\left (\frac {1}{2}-\frac {a_{0} b_{2} -a_{2} b_{0}}{4 a_{0} \left (a_{2} x +b_{2} \right )}\right ) \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )+\frac {i a_{2}^{{3}/{2}} \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )}{2 \sqrt {a_{0}}\, \sqrt {\lambda }\, \left (a_{2} x +b_{2} \right )}\right ) \sqrt {a_{0}}\, \sqrt {\lambda }}{\sqrt {a_{2}}}}{\left (-\frac {a_{2} \lambda x}{a_{2} x +b_{2}}-\frac {b_{2} \lambda }{a_{2} x +b_{2}}\right ) \left (\operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )+c_3 \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )\right )} \]

Simplifying the above gives

\begin{align*} y &= \frac {\left (a_{2}^{{3}/{2}} a_{0}^{{3}/{2}}+\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }\, a_{0}}{2}\right ) \operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )-a_{2}^{{3}/{2}} a_{0}^{{3}/{2}} \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right ) c_3 +i \sqrt {\lambda }\, \left (\left (a_{2} x +\frac {b_{2}}{2}\right ) a_{0} +\frac {a_{2} b_{0}}{2}\right ) a_{0} \left (\operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )+c_3 \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )\right )}{\sqrt {a_{2}}\, a_{0}^{{3}/{2}} \lambda \left (a_{2} x +b_{2} \right ) \left (\operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )+c_3 \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )\right )} \\ \end{align*}

Summary of solutions found

\begin{align*} y &= \frac {\left (a_{2}^{{3}/{2}} a_{0}^{{3}/{2}}+\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }\, a_{0}}{2}\right ) \operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )-a_{2}^{{3}/{2}} a_{0}^{{3}/{2}} \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right ) c_3 +i \sqrt {\lambda }\, \left (\left (a_{2} x +\frac {b_{2}}{2}\right ) a_{0} +\frac {a_{2} b_{0}}{2}\right ) a_{0} \left (\operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )+c_3 \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )\right )}{\sqrt {a_{2}}\, a_{0}^{{3}/{2}} \lambda \left (a_{2} x +b_{2} \right ) \left (\operatorname {WhittakerM}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )+c_3 \operatorname {WhittakerW}\left (\frac {i \left (a_{0} b_{2} -a_{2} b_{0} \right ) \sqrt {\lambda }}{2 a_{2}^{{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}^{{3}/{2}}}\right )\right )} \\ \end{align*}

2.2.11.2 ✓ Maple. Time used: 0.017 (sec). Leaf size: 461

ode:=(a__2*x+b__2)*(diff(y(x),x)+lambda*y(x)^2)+a__0*x+b__0 = 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: 
   <- Abel AIR successful: ODE belongs to the 1F1 2-parameter class
 

Maple step by step

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

2.2.11.3 ✓ Mathematica. Time used: 0.888 (sec). Leaf size: 540

ode=(a2*x+b2)*(D[y[x],x]+\[Lambda]*y[x]^2)+a0*x+b0==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

2.2.11.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a__0 = symbols("a__0") 
a__2 = symbols("a__2") 
b__0 = symbols("b__0") 
b__2 = symbols("b__2") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(a__0*x + b__0 + (a__2*x + b__2)*(lambda_*y(x)**2 + Derivative(y(x), x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out