2.2.22 Problem 24

Internal problem ID [13228]
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 : 24
Date solved : Wednesday, December 31, 2025 at 12:15:47 PM
CAS classification : [_Riccati]

2.2.22.1 Solved using first_order_ode_riccati

1.534 (sec)

Entering first order ode riccati solver

\begin{align*} y^{\prime }&=a y^{2}+y b +c x +k \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= a y^{2}+y b +c x +k \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 \(f_0(x)=c x +k\), \(f_1(x)=b\) and \(f_2(x)=a\). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u a} \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 &=a b\\ f_2^2 f_0 &=a^{2} \left (c x +k \right ) \end{align*}

Substituting the above terms back in equation (2) gives

\[ a u^{\prime \prime }\left (x \right )-a b u^{\prime }\left (x \right )+a^{2} \left (c x +k \right ) u \left (x \right ) = 0 \]

Entering second order Airy solverThis is Airy ODE. It has the general form

\[ a \frac {d^{2}u}{d x^{2}} + b \frac {d u}{d x} + c u x = F(x) \]

Where in this case

\begin{align*} a &= a\\ b &= -a b\\ c &= \frac {a^{2} \left (c x +k \right )}{x}\\ F &= 0 \end{align*}

Therefore the solution to the homogeneous Airy ODE becomes

\[ u = c_1 \,{\mathrm e}^{\frac {b x}{2}} \operatorname {AiryAi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )+c_2 \,{\mathrm e}^{\frac {b x}{2}} \operatorname {AiryBi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right ) \]

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = \frac {c_1 b \,{\mathrm e}^{\frac {b x}{2}} \operatorname {AiryAi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )}{2}-c_1 \,{\mathrm e}^{\frac {b x}{2}} \left (a c \right )^{{1}/{3}} \operatorname {AiryAi}\left (1, -\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )+\frac {c_2 b \,{\mathrm e}^{\frac {b x}{2}} \operatorname {AiryBi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )}{2}-c_2 \,{\mathrm e}^{\frac {b x}{2}} \left (a c \right )^{{1}/{3}} \operatorname {AiryBi}\left (1, -\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right ) \end{equation}

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

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u a} \\ y &= -\frac {\frac {c_1 b \,{\mathrm e}^{\frac {b x}{2}} \operatorname {AiryAi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )}{2}-c_1 \,{\mathrm e}^{\frac {b x}{2}} \left (a c \right )^{{1}/{3}} \operatorname {AiryAi}\left (1, -\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )+\frac {c_2 b \,{\mathrm e}^{\frac {b x}{2}} \operatorname {AiryBi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )}{2}-c_2 \,{\mathrm e}^{\frac {b x}{2}} \left (a c \right )^{{1}/{3}} \operatorname {AiryBi}\left (1, -\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )}{a \left (c_1 \,{\mathrm e}^{\frac {b x}{2}} \operatorname {AiryAi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )+c_2 \,{\mathrm e}^{\frac {b x}{2}} \operatorname {AiryBi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )\right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {\frac {b \,{\mathrm e}^{\frac {b x}{2}} \operatorname {AiryAi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )}{2}-{\mathrm e}^{\frac {b x}{2}} \left (a c \right )^{{1}/{3}} \operatorname {AiryAi}\left (1, -\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )+\frac {c_3 b \,{\mathrm e}^{\frac {b x}{2}} \operatorname {AiryBi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )}{2}-c_3 \,{\mathrm e}^{\frac {b x}{2}} \left (a c \right )^{{1}/{3}} \operatorname {AiryBi}\left (1, -\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )}{a \left ({\mathrm e}^{\frac {b x}{2}} \operatorname {AiryAi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )+c_3 \,{\mathrm e}^{\frac {b x}{2}} \operatorname {AiryBi}\left (-\frac {\left (a^{3} \left (c x +k \right )-\frac {a^{2} b^{2}}{4}\right ) \left (a c \right )^{{1}/{3}}}{c \,a^{3}}\right )\right )} \]

Simplifying the above gives

\begin{align*} y &= \frac {-b \operatorname {AiryAi}\left (-\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right )-c_3 b \operatorname {AiryBi}\left (-\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right )+2 \left (a c \right )^{{1}/{3}} \left (\operatorname {AiryBi}\left (1, -\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right ) c_3 +\operatorname {AiryAi}\left (1, -\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right )\right )}{2 a \left (\operatorname {AiryBi}\left (-\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right ) c_3 +\operatorname {AiryAi}\left (-\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right )\right )} \\ \end{align*}

Summary of solutions found

\begin{align*} y &= \frac {-b \operatorname {AiryAi}\left (-\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right )-c_3 b \operatorname {AiryBi}\left (-\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right )+2 \left (a c \right )^{{1}/{3}} \left (\operatorname {AiryBi}\left (1, -\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right ) c_3 +\operatorname {AiryAi}\left (1, -\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right )\right )}{2 a \left (\operatorname {AiryBi}\left (-\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right ) c_3 +\operatorname {AiryAi}\left (-\frac {\left (a c \right )^{{1}/{3}} \left (\left (c x +k \right ) a -\frac {b^{2}}{4}\right )}{a c}\right )\right )} \\ \end{align*}

2.2.22.2 ✓ Maple. Time used: 0.009 (sec). Leaf size: 194

ode:=diff(y(x),x) = a*y(x)^2+b*y(x)+c*x+k; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {2 \left (\frac {c}{\sqrt {a}}\right )^{{1}/{3}} \sqrt {a}\, \left (\operatorname {AiryAi}\left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right ) c_1 +\operatorname {AiryBi}\left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )\right )-\left (c_1 \operatorname {AiryAi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )+\operatorname {AiryBi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )\right ) b}{2 a \left (c_1 \operatorname {AiryAi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )+\operatorname {AiryBi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )\right )} \]

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 0F1 0-parameter (Airy type) class
 

Maple step by step

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

2.2.22.3 ✓ Mathematica. Time used: 0.175 (sec). Leaf size: 284

ode=D[y[x],x]==a*y[x]^2+b*y[x]+c*x+k; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {c \left (-b (-a c)^{2/3} \operatorname {AiryBi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+2 a c \operatorname {AiryBiPrime}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+c_1 \left (2 a c \operatorname {AiryAiPrime}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )-b (-a c)^{2/3} \operatorname {AiryAi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )\right )\right )}{2 (-a c)^{5/3} \left (\operatorname {AiryBi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )+c_1 \operatorname {AiryAi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )\right )}\\ y(x)&\to -\frac {\frac {2 \sqrt [3]{-a c} \operatorname {AiryAiPrime}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )}{\operatorname {AiryAi}\left (\frac {b^2-4 a (k+c x)}{4 (-a c)^{2/3}}\right )}+b}{2 a} \end{align*}

2.2.22.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
k = symbols("k") 
y = Function("y") 
ode = Eq(-a*y(x)**2 - b*y(x) - c*x - k + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*y(x)**2 - b*y(x) - c*x - k + Derivative(y(x), x) cannot be solved by the lie group method