61.24.5 problem 5
Internal
problem
ID
[12418]
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
1.3.3-2.
Problem
number
:
5
Date
solved
:
Tuesday, January 28, 2025 at 07:58:24 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime } y+x \left (a \,x^{2}+b \right ) y+x&=0 \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 179
dsolve(y(x)*diff(y(x),x)+x*(a*x^2+b)*y(x)+x=0,y(x), singsol=all)
\[
\frac {2 \operatorname {AiryBi}\left (1, \frac {4 a y+\left (a \,x^{2}+b \right )^{2}}{4 a^{{2}/{3}}}\right ) a^{{1}/{3}} c_{1} +c_{1} \left (a \,x^{2}+b \right ) \operatorname {AiryBi}\left (\frac {4 a y+\left (a \,x^{2}+b \right )^{2}}{4 a^{{2}/{3}}}\right )-2 \operatorname {AiryAi}\left (1, \frac {4 a y+\left (a \,x^{2}+b \right )^{2}}{4 a^{{2}/{3}}}\right ) a^{{1}/{3}}-\left (a \,x^{2}+b \right ) \operatorname {AiryAi}\left (\frac {4 a y+\left (a \,x^{2}+b \right )^{2}}{4 a^{{2}/{3}}}\right )}{2 \operatorname {AiryBi}\left (1, \frac {4 a y+\left (a \,x^{2}+b \right )^{2}}{4 a^{{2}/{3}}}\right ) a^{{1}/{3}}+\operatorname {AiryBi}\left (\frac {4 a y+\left (a \,x^{2}+b \right )^{2}}{4 a^{{2}/{3}}}\right ) \left (a \,x^{2}+b \right )} = 0
\]
✓ Solution by Mathematica
Time used: 0.314 (sec). Leaf size: 143
DSolve[y[x]*D[y[x],x]+x*(a*x^2+b)*y[x]+x==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\frac {\left (a x^2+b\right ) \operatorname {AiryAi}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )+2 \sqrt [3]{a} \operatorname {AiryAiPrime}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )}{\left (a x^2+b\right ) \operatorname {AiryBi}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )+2 \sqrt [3]{a} \operatorname {AiryBiPrime}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )}+c_1=0,y(x)\right ]
\]