61.23.1 problem 1
Internal
problem
ID
[12323]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.3.
Abel
Equations
of
the
Second
Kind.
subsection
1.3.2.
Problem
number
:
1
Date
solved
:
Monday, March 31, 2025 at 05:10:03 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y y^{\prime }&=\left (a x +b \right ) y+1 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 215
ode:=y(x)*diff(y(x),x) = (a*x+b)*y(x)+1;
dsolve(ode,y(x), singsol=all);
\[
\frac {-\left (-\operatorname {AiryBi}\left (-\frac {\left (-2 y a +\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right ) c_1 +\operatorname {AiryAi}\left (-\frac {\left (-2 y a +\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right )\right ) \left (-a^{2}\right )^{{1}/{3}} \left (a x +b \right ) 2^{{1}/{3}}+2 a \left (\operatorname {AiryBi}\left (1, -\frac {\left (-2 y a +\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right ) c_1 -\operatorname {AiryAi}\left (1, -\frac {\left (-2 y a +\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right )\right )}{2^{{1}/{3}} \left (-a^{2}\right )^{{1}/{3}} \left (a x +b \right ) \operatorname {AiryBi}\left (-\frac {\left (-2 y a +\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right )+2 \operatorname {AiryBi}\left (1, -\frac {\left (-2 y a +\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right ) a} = 0
\]
✓ Mathematica. Time used: 0.539 (sec). Leaf size: 161
ode=y[x]*D[y[x],x]==(a*x+b)*y[x]+1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\sqrt [3]{2} (a x+b) \operatorname {AiryAi}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )-2 \sqrt [3]{a} \operatorname {AiryAiPrime}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )}{\sqrt [3]{2} (a x+b) \operatorname {AiryBi}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )-2 \sqrt [3]{a} \operatorname {AiryBiPrime}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )}+c_1=0,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq((-a*x - b)*y(x) + y(x)*Derivative(y(x), x) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a*x - b + Derivative(y(x), x) - 1/y(x) cannot be solved by the factorable group method