Internal problem ID [9939]
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. Form \(y y'-y=f(x)\). subsection 1.3.1-2. Solvable
equations and their solutions
Problem number: 33.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime } y-y-\frac {A}{x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 209
dsolve(y(x)*diff(y(x),x)-y(x)=A*x^(-2),y(x), singsol=all)
\[ c_{1}+\frac {-A 2^{\frac {1}{3}} \left (x -y \relax (x )\right ) \AiryAi \left (\frac {\left (-x^{3}+2 x^{2} y \relax (x )-y \relax (x )^{2} x -2 A \right ) 2^{\frac {2}{3}}}{4 \left (-A^{2}\right )^{\frac {1}{3}} x}\right )+2 \AiryAi \left (1, \frac {\left (-x^{3}+2 x^{2} y \relax (x )-y \relax (x )^{2} x -2 A \right ) 2^{\frac {2}{3}}}{4 \left (-A^{2}\right )^{\frac {1}{3}} x}\right ) \left (-A^{2}\right )^{\frac {2}{3}}}{A 2^{\frac {1}{3}} \left (x -y \relax (x )\right ) \AiryBi \left (\frac {\left (-x^{3}+2 x^{2} y \relax (x )-y \relax (x )^{2} x -2 A \right ) 2^{\frac {2}{3}}}{4 \left (-A^{2}\right )^{\frac {1}{3}} x}\right )-2 \AiryBi \left (1, \frac {\left (-x^{3}+2 x^{2} y \relax (x )-y \relax (x )^{2} x -2 A \right ) 2^{\frac {2}{3}}}{4 \left (-A^{2}\right )^{\frac {1}{3}} x}\right ) \left (-A^{2}\right )^{\frac {2}{3}}} = 0 \]
✓ Solution by Mathematica
Time used: 0.602 (sec). Leaf size: 201
DSolve[y[x]*y'[x]-y[x]==A*x^(-2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\text {Ai}'\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )-\frac {(x-y(x)) \text {Ai}\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )}{2^{2/3} \sqrt [3]{A}}}{\text {Bi}'\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )-\frac {(x-y(x)) \text {Bi}\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )}{2^{2/3} \sqrt [3]{A}}}+c_1=0,y(x)\right ] \]