55.23.2 problem 2
Internal
problem
ID
[13619]
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
:
2
Date
solved
:
Sunday, October 12, 2025 at 04:13:34 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y y^{\prime }&=\frac {y}{\left (a x +b \right )^{2}}+1 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 557
ode:=y(x)*diff(y(x),x) = 1/(a*x+b)^2*y(x)+1;
dsolve(ode,y(x), singsol=all);
\[
\frac {\left (1+a \left (a x +b \right ) y\right ) \left (\operatorname {AiryBi}\left (-\frac {2^{{2}/{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y^{2}}{2}+\left (-a^{2} x -a b \right ) y+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{{1}/{3}} \left (a x +b \right )^{2}}\right ) c_1 -\operatorname {AiryAi}\left (-\frac {2^{{2}/{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y^{2}}{2}+\left (-a^{2} x -a b \right ) y+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{{1}/{3}} \left (a x +b \right )^{2}}\right )\right ) a 2^{{1}/{3}}+2 \left (a x +b \right ) \left (a^{2}\right )^{{2}/{3}} \left (\operatorname {AiryBi}\left (1, -\frac {2^{{2}/{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y^{2}}{2}+\left (-a^{2} x -a b \right ) y+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{{1}/{3}} \left (a x +b \right )^{2}}\right ) c_1 -\operatorname {AiryAi}\left (1, -\frac {2^{{2}/{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y^{2}}{2}+\left (-a^{2} x -a b \right ) y+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{{1}/{3}} \left (a x +b \right )^{2}}\right )\right )}{\left (1+a \left (a x +b \right ) y\right ) 2^{{1}/{3}} a \operatorname {AiryBi}\left (-\frac {2^{{2}/{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y^{2}}{2}+\left (-a^{2} x -a b \right ) y+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{{1}/{3}} \left (a x +b \right )^{2}}\right )+2 \operatorname {AiryBi}\left (1, -\frac {2^{{2}/{3}} \left (-\frac {a^{2} \left (a x +b \right )^{2} y^{2}}{2}+\left (-a^{2} x -a b \right ) y+a^{4} x^{3}+3 a^{3} b \,x^{2}+3 a^{2} b^{2} x +a \,b^{3}-\frac {1}{2}\right )}{2 \left (a^{2}\right )^{{1}/{3}} \left (a x +b \right )^{2}}\right ) \left (a^{2}\right )^{{2}/{3}} \left (a x +b \right )} = 0
\]
✓ Mathematica. Time used: 0.803 (sec). Leaf size: 561
ode=y[x]*D[y[x],x]==(a*x+b)^(-2)*y[x]+1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {a y(x) (a x+b) \operatorname {AiryAi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\operatorname {AiryAi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \operatorname {AiryAiPrime}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}{a y(x) (a x+b) \operatorname {AiryBi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\operatorname {AiryBi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \operatorname {AiryBiPrime}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}+c_1=0,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(y(x)*Derivative(y(x), x) - 1 - y(x)/(a*x + b)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out