23.1.62 problem 56

Internal problem ID [4669]
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 : 56
Date solved : Tuesday, September 30, 2025 at 07:52:45 AM
CAS classification : [[_Riccati, _special]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 59

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

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

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 331

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

\begin{align*} y(x)&\to \frac {\sqrt {a} \sqrt {b} x^{3/2} \left (-2 \operatorname {BesselJ}\left (-\frac {2}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )+c_1 \left (\operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )-\operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )\right )\right )-c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )}{2 b x \left (\operatorname {BesselJ}\left (\frac {1}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )+c_1 \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )\right )}\\ y(x)&\to -\frac {\sqrt {a} \sqrt {b} x^{3/2} \operatorname {BesselJ}\left (-\frac {4}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )-\sqrt {a} \sqrt {b} x^{3/2} \operatorname {BesselJ}\left (\frac {2}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )+\operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )}{2 b x \operatorname {BesselJ}\left (-\frac {1}{3},\frac {2}{3} \sqrt {a} \sqrt {b} x^{3/2}\right )} \end{align*}

✗ Sympy

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

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