Internal problem ID [10010]
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 1.3.3-2. Equations of
the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 16.
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 }-a \left (1-\frac {b}{\sqrt {x}}\right ) y-\frac {a^{2} b}{\sqrt {x}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 183
dsolve(y(x)*diff(y(x),x)-a*(1-b*x^(-1/2))*y(x)=a^2*b*x^(-1/2),y(x), singsol=all)
\[ c_{1}+\frac {-2^{\frac {2}{3}} \left (b^{2}\right )^{\frac {1}{3}} \left (-\sqrt {x}+b \right ) \AiryAi \left (\frac {\left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y \relax (x )\right ) 2^{\frac {1}{3}}}{2 \left (b^{2}\right )^{\frac {1}{3}} a}\right )-2 \AiryAi \left (1, \frac {\left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y \relax (x )\right ) 2^{\frac {1}{3}}}{2 \left (b^{2}\right )^{\frac {1}{3}} a}\right ) b}{2^{\frac {2}{3}} \left (b^{2}\right )^{\frac {1}{3}} \left (-\sqrt {x}+b \right ) \AiryBi \left (\frac {\left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y \relax (x )\right ) 2^{\frac {1}{3}}}{2 \left (b^{2}\right )^{\frac {1}{3}} a}\right )+2 \AiryBi \left (1, \frac {\left (-2 \sqrt {x}\, a b +\left (b^{2}+x \right ) a -y \relax (x )\right ) 2^{\frac {1}{3}}}{2 \left (b^{2}\right )^{\frac {1}{3}} a}\right ) b} = 0 \]
✓ Solution by Mathematica
Time used: 1.803 (sec). Leaf size: 323
DSolve[y[x]*y'[x]-a*(1-b*x^(-1/2))*y[x]==a^2*b*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\sqrt [3]{-1} 2^{2/3} \sqrt [3]{\left (b-\sqrt {x}\right )^3} \text {Ai}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )-2 \sqrt [3]{b} \text {Ai}'\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )}{\sqrt [3]{-1} 2^{2/3} \sqrt [3]{\left (b-\sqrt {x}\right )^3} \text {Bi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )-2 \sqrt [3]{b} \text {Bi}'\left (\frac {\left (-\frac {1}{2}\right )^{2/3} \left (\left (b-\sqrt {x}\right )^3\right )^{2/3} \left (a \left (b-\sqrt {x}\right )^2-y(x)\right )}{a b^{2/3} \left (b-\sqrt {x}\right )^2}\right )}+c_1=0,y(x)\right ] \]