Internal problem ID [9938]
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: 32.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y y^{\prime }-y-\frac {A}{\sqrt {x}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 159
dsolve(y(x)*diff(y(x),x)-y(x)=A*x^(-1/2),y(x), singsol=all)
\[ c_{1}+\frac {-2^{\frac {2}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {1}{3}} \AiryAi \left (\frac {2^{\frac {1}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (-y \relax (x )+x \right )}{2 A^{2} x}\right )-2 \AiryAi \left (1, \frac {2^{\frac {1}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (-y \relax (x )+x \right )}{2 A^{2} x}\right ) A}{2^{\frac {2}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {1}{3}} \AiryBi \left (\frac {2^{\frac {1}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (-y \relax (x )+x \right )}{2 A^{2} x}\right )+2 \AiryBi \left (1, \frac {2^{\frac {1}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (-y \relax (x )+x \right )}{2 A^{2} x}\right ) A} = 0 \]
✓ Solution by Mathematica
Time used: 0.311 (sec). Leaf size: 139
DSolve[y[x]*y'[x]-y[x]==A*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\sqrt [3]{-1} 2^{2/3} \sqrt {x} \text {AiryAi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )+2 \sqrt [3]{A} \text {AiryAiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )}{\sqrt [3]{-1} 2^{2/3} \sqrt {x} \text {Bi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )+2 \sqrt [3]{A} \text {Bi}'\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )}+c_1=0,y(x)\right ] \]