23.1.60 problem 54

Internal problem ID [4667]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 54
Date solved : Tuesday, September 30, 2025 at 07:50:51 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=3 a +3 b x +3 b y^{2} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 80

ode:=diff(y(x),x) = 3*a+3*b*x+3*b*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.142 (sec). Leaf size: 191

ode=D[y[x],x]==3*(a+b*x+ b*y[x]^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-3*a - 3*b*x - 3*b*y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -3*a - 3*b*x - 3*b*y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method