61.23.1 problem 1
Internal
problem
ID
[12402]
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
:
Tuesday, January 28, 2025 at 07:58:06 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime } y&=\left (a x +b \right ) y+1 \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 214
dsolve(y(x)*diff(y(x),x)=(a*x+b)*y(x)+1,y(x), singsol=all)
\[
\frac {-\left (-\operatorname {AiryBi}\left (-\frac {\left (-2 a y+\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right ) c_{1} +\operatorname {AiryAi}\left (-\frac {\left (-2 a y+\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right )\right ) \left (a x +b \right ) \left (-a^{2}\right )^{{1}/{3}} 2^{{1}/{3}}-2 a \left (-\operatorname {AiryBi}\left (1, -\frac {\left (-2 a y+\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right ) c_{1} +\operatorname {AiryAi}\left (1, -\frac {\left (-2 a y+\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right )\right )}{2^{{1}/{3}} \left (-a^{2}\right )^{{1}/{3}} \left (a x +b \right ) \operatorname {AiryBi}\left (-\frac {\left (-2 a y+\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right )+2 \operatorname {AiryBi}\left (1, -\frac {\left (-2 a y+\left (a x +b \right )^{2}\right ) 2^{{2}/{3}}}{4 \left (-a^{2}\right )^{{1}/{3}}}\right ) a} = 0
\]
✓ Solution by Mathematica
Time used: 0.539 (sec). Leaf size: 161
DSolve[y[x]*D[y[x],x]==(a*x+b)*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\frac {\sqrt [3]{2} (a x+b) \operatorname {AiryAi}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )-2 \sqrt [3]{a} \operatorname {AiryAiPrime}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )}{\sqrt [3]{2} (a x+b) \operatorname {AiryBi}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )-2 \sqrt [3]{a} \operatorname {AiryBiPrime}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )}+c_1=0,y(x)\right ]
\]