61.2.24 problem 24
Internal
problem
ID
[11951]
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
:
Sunday, March 30, 2025 at 09:26:13 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a y^{2}+b y+c x +k \end{align*}
✓ Maple. Time used: 0.011 (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);
\[
y = \frac {2 \left (\frac {c}{\sqrt {a}}\right )^{{1}/{3}} \sqrt {a}\, \left (\operatorname {AiryAi}\left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right ) c_1 +\operatorname {AiryBi}\left (1, -\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )\right )-b \left (c_1 \operatorname {AiryAi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )+\operatorname {AiryBi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )\right )}{2 a \left (c_1 \operatorname {AiryAi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )+\operatorname {AiryBi}\left (-\frac {a \left (c x +k \right )-\frac {b^{2}}{4}}{\left (\frac {c}{\sqrt {a}}\right )^{{2}/{3}} a}\right )\right )}
\]
✓ Mathematica. Time used: 0.271 (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*}
✗ 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