61.23.2 problem 2
Internal
problem
ID
[12403]
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
:
Tuesday, January 28, 2025 at 07:58:07 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y^{\prime } y&=\frac {y}{\left (a x +b \right )^{2}}+1 \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 556
dsolve(y(x)*diff(y(x),x)=(a*x+b)^(-2)*y(x)+1,y(x), singsol=all)
\[
\frac {-a \left (1+a \left (a x +b \right ) y\right ) \left (-\operatorname {AiryBi}\left (-\frac {\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} x \,b^{2}+b^{3} a -\frac {1}{2}\right ) 2^{{2}/{3}}}{2 \left (a^{2}\right )^{{1}/{3}} \left (a x +b \right )^{2}}\right ) c_{1} +\operatorname {AiryAi}\left (-\frac {\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} x \,b^{2}+b^{3} a -\frac {1}{2}\right ) 2^{{2}/{3}}}{2 \left (a^{2}\right )^{{1}/{3}} \left (a x +b \right )^{2}}\right )\right ) 2^{{1}/{3}}-2 \left (a^{2}\right )^{{2}/{3}} \left (a x +b \right ) \left (-\operatorname {AiryBi}\left (1, -\frac {\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} x \,b^{2}+b^{3} a -\frac {1}{2}\right ) 2^{{2}/{3}}}{2 \left (a^{2}\right )^{{1}/{3}} \left (a x +b \right )^{2}}\right ) c_{1} +\operatorname {AiryAi}\left (1, -\frac {\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} x \,b^{2}+b^{3} a -\frac {1}{2}\right ) 2^{{2}/{3}}}{2 \left (a^{2}\right )^{{1}/{3}} \left (a x +b \right )^{2}}\right )\right )}{a \left (1+a \left (a x +b \right ) y\right ) 2^{{1}/{3}} \operatorname {AiryBi}\left (-\frac {\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} x \,b^{2}+b^{3} a -\frac {1}{2}\right ) 2^{{2}/{3}}}{2 \left (a^{2}\right )^{{1}/{3}} \left (a x +b \right )^{2}}\right )+2 \operatorname {AiryBi}\left (1, -\frac {\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} x \,b^{2}+b^{3} a -\frac {1}{2}\right ) 2^{{2}/{3}}}{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
\]
✓ Solution by Mathematica
Time used: 1.367 (sec). Leaf size: 561
DSolve[y[x]*D[y[x],x]==(a*x+b)^(-2)*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\frac {a y(x) (a x+b) \operatorname {AiryAi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\operatorname {AiryAi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \operatorname {AiryAiPrime}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}{a y(x) (a x+b) \operatorname {AiryBi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\operatorname {AiryBi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \operatorname {AiryBiPrime}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}+c_1=0,y(x)\right ]
\]