29.3.2 problem 56

Internal problem ID [4664]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 3
Problem number : 56
Date solved : Tuesday, March 04, 2025 at 07:01:41 PM
CAS classification : [_Riccati]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 85

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

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

✓ Mathematica. Time used: 0.203 (sec). Leaf size: 143

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

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

✗ Sympy

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

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