Internal problem ID [9917]
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: 11.
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 {2 x}{9}-6 A^{2} \left (1+\frac {2 A}{\sqrt {x}}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 628
dsolve(y(x)*diff(y(x),x)-y(x)=-2/9*x+6*A^2*(1+2*A*x^(-1/2)),y(x), singsol=all)
\[ -\frac {108 A^{3} \ln \left (-\frac {108 A^{3}+54 A^{2} \sqrt {x}-2 x^{\frac {3}{2}}+9 A y \relax (x )+3 y \relax (x ) \sqrt {x}}{3 A y \relax (x )}\right )-108 A^{3} \ln \left (-\frac {108 A^{3}+54 A^{2} \sqrt {x}-2 x^{\frac {3}{2}}-18 A y \relax (x )+3 y \relax (x ) \sqrt {x}}{6 A y \relax (x )}\right )+54 A^{2} \ln \left (-\frac {108 A^{3}+54 A^{2} \sqrt {x}-2 x^{\frac {3}{2}}+9 A y \relax (x )+3 y \relax (x ) \sqrt {x}}{3 A y \relax (x )}\right ) \sqrt {x}-54 A^{2} \ln \left (-\frac {108 A^{3}+54 A^{2} \sqrt {x}-2 x^{\frac {3}{2}}-18 A y \relax (x )+3 y \relax (x ) \sqrt {x}}{6 A y \relax (x )}\right ) \sqrt {x}-2 \ln \left (-\frac {108 A^{3}+54 A^{2} \sqrt {x}-2 x^{\frac {3}{2}}+9 A y \relax (x )+3 y \relax (x ) \sqrt {x}}{3 A y \relax (x )}\right ) x^{\frac {3}{2}}+2 \ln \left (-\frac {108 A^{3}+54 A^{2} \sqrt {x}-2 x^{\frac {3}{2}}-18 A y \relax (x )+3 y \relax (x ) \sqrt {x}}{6 A y \relax (x )}\right ) x^{\frac {3}{2}}+9 A \ln \left (-\frac {108 A^{3}+54 A^{2} \sqrt {x}-2 x^{\frac {3}{2}}+9 A y \relax (x )+3 y \relax (x ) \sqrt {x}}{3 A y \relax (x )}\right ) y \relax (x )-9 A \ln \left (-\frac {108 A^{3}+54 A^{2} \sqrt {x}-2 x^{\frac {3}{2}}-18 A y \relax (x )+3 y \relax (x ) \sqrt {x}}{6 A y \relax (x )}\right ) y \relax (x )+3 \ln \left (-\frac {108 A^{3}+54 A^{2} \sqrt {x}-2 x^{\frac {3}{2}}+9 A y \relax (x )+3 y \relax (x ) \sqrt {x}}{3 A y \relax (x )}\right ) \sqrt {x}\, y \relax (x )-3 \ln \left (-\frac {108 A^{3}+54 A^{2} \sqrt {x}-2 x^{\frac {3}{2}}-18 A y \relax (x )+3 y \relax (x ) \sqrt {x}}{6 A y \relax (x )}\right ) \sqrt {x}\, y \relax (x )-27 A y \relax (x )}{3 \left (108 A^{3}+54 A^{2} \sqrt {x}-2 x^{\frac {3}{2}}+9 A y \relax (x )+3 y \relax (x ) \sqrt {x}\right )}-\frac {9 A^{2}}{9 A^{2}-x}+\frac {\ln \left (9 A^{2}-x \right )}{6}-\frac {\ln \left (36 A^{2}-x \right )}{6}+\frac {\ln \left (3 A +\sqrt {x}\right )}{6}-\frac {3 A}{2 \left (3 A +\sqrt {x}\right )}+\frac {\ln \left (6 A +\sqrt {x}\right )}{6}-\frac {\ln \left (3 A -\sqrt {x}\right )}{6}+\frac {3 A}{2 \left (3 A -\sqrt {x}\right )}-\frac {\ln \left (6 A -\sqrt {x}\right )}{6}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 7.8 (sec). Leaf size: 488
DSolve[y[x]*y'[x]-y[x]==-2/9*x+6*A^2*(1+2*A*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {2^{2/3} \left (\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{\sqrt [3]{A^3}}+54\right ) \left (\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+9 \sqrt {x} y(x)}{\sqrt [3]{A^3} y(x)}+27\right ) \left (-\frac {\left (3 \left (3 \sqrt [3]{A^3}+\sqrt {x}\right ) y(x)+2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2\right ) \log \left (\frac {1}{27} 2^{2/3} \left (\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{\sqrt [3]{A^3}}+54\right )\right )}{9 \sqrt [3]{A^3} y(x)}+\left (\frac {2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+3 \sqrt {x} y(x)}{9 \sqrt [3]{A^3} y(x)}+1\right ) \log \left (\frac {1}{27} 2^{2/3} \left (\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+9 \sqrt {x} y(x)}{\sqrt [3]{A^3} y(x)}+27\right )\right )-3\right )}{6561 \left (\frac {\left (2 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2+3 \sqrt {x} y(x)\right )^3}{729 A^3 y(x)^3}+\frac {-\frac {6 \left (6 A-\sqrt {x}\right ) \left (3 A+\sqrt {x}\right )^2}{y(x)}-9 \sqrt {x}}{9 \sqrt [3]{A^3}}-2\right )}=\frac {2^{2/3} \left (A^3\right )^{2/3} \left (\frac {9 A}{3 A+\sqrt {x}}+2 \tanh ^{-1}\left (\frac {1}{3}-\frac {2 \sqrt {x}}{9 A}\right )\right )}{9 A^2}+c_1,y(x)\right ] \]