61.22.33 problem 33
Internal
problem
ID
[12358]
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
1.3.1-2.
Solvable
equations
and
their
solutions
Problem
number
:
33
Date
solved
:
Tuesday, January 28, 2025 at 07:56:17 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y^{\prime } y-y&=\frac {A}{x^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 276
dsolve(y(x)*diff(y(x),x)-y(x)=A*x^(-2),y(x), singsol=all)
\[
\frac {A \left (x -y\right ) \left (-\operatorname {AiryBi}\left (-\frac {\left (x^{3}-2 x^{2} y+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right ) c_{1} +\operatorname {AiryAi}\left (-\frac {\left (x^{3}-2 x^{2} y+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right )\right ) 2^{{1}/{3}}-2 \left (-\operatorname {AiryBi}\left (1, -\frac {\left (x^{3}-2 x^{2} y+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right ) c_{1} +\operatorname {AiryAi}\left (1, -\frac {\left (x^{3}-2 x^{2} y+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right )\right ) \left (-A^{2}\right )^{{2}/{3}}}{-A 2^{{1}/{3}} \left (x -y\right ) \operatorname {AiryBi}\left (-\frac {\left (x^{3}-2 x^{2} y+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right )+2 \operatorname {AiryBi}\left (1, -\frac {\left (x^{3}-2 x^{2} y+x y^{2}+2 A \right ) 2^{{2}/{3}}}{4 \left (-A^{2}\right )^{{1}/{3}} x}\right ) \left (-A^{2}\right )^{{2}/{3}}} = 0
\]
✓ Solution by Mathematica
Time used: 0.574 (sec). Leaf size: 201
DSolve[y[x]*D[y[x],x]-y[x]==A*x^(-2),y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\frac {\operatorname {AiryAiPrime}\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )-\frac {(x-y(x)) \operatorname {AiryAi}\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )}{2^{2/3} \sqrt [3]{A}}}{\operatorname {AiryBiPrime}\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )-\frac {(x-y(x)) \operatorname {AiryBi}\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )}{2^{2/3} \sqrt [3]{A}}}+c_1=0,y(x)\right ]
\]