Internal problem ID [9932]
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: 26.
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}-\frac {A}{\sqrt {x}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 152
dsolve(y(x)*diff(y(x),x)-y(x)=-2/9*x+A*x^(-1/2),y(x), singsol=all)
\[ y \relax (x ) = -\frac {2 \,6^{\frac {1}{3}} \sqrt {3}\, \left (-2 x^{\frac {3}{2}}+9 A \right )}{3 \sqrt {x}\, \left (3 \sqrt {3}\, \left (\frac {A}{x^{\frac {3}{2}}}\right )^{\frac {1}{3}}-9 \left (\frac {A}{x^{\frac {3}{2}}}\right )^{\frac {1}{3}} \tan \left (\RootOf \left (-18 \sqrt {3}\, 6^{\frac {1}{3}} \left (\int \frac {\left (\frac {A}{x^{\frac {3}{2}}}\right )^{\frac {2}{3}} \sqrt {x}}{-2 x^{\frac {3}{2}}+9 A}d x \right )-\sqrt {3}\, \ln \left (-\frac {4 \sqrt {3}\, \left (\tan ^{3}\left (\textit {\_Z} \right )\right )-\left (\tan ^{4}\left (\textit {\_Z} \right )\right )+12 \sqrt {3}\, \tan \left (\textit {\_Z} \right )-18 \left (\tan ^{2}\left (\textit {\_Z} \right )\right )-9}{\tan ^{4}\left (\textit {\_Z} \right )+2 \left (\tan ^{2}\left (\textit {\_Z} \right )\right )+1}\right )+12 \sqrt {3}\, c_{1}+12 \textit {\_Z} \right )\right )+2 \,6^{\frac {1}{3}} \sqrt {3}\right )} \]
✓ Solution by Mathematica
Time used: 0.77 (sec). Leaf size: 282
DSolve[y[x]*y'[x]-y[x]==-2/9*x+A*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\log \left (9 A^{2/3}+3 \sqrt [3]{6} \sqrt [3]{A} \sqrt {x}+6^{2/3} x\right )+2 \sqrt {3} \text {ArcTan}\left (\frac {-\frac {6 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}-27}{27 \sqrt {3}}\right )+2 \sqrt {3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{6} \sqrt {x}}{\sqrt [3]{A}}+3}{3 \sqrt {3}}\right )+2 \log \left (\frac {1}{27} \left (27-\frac {3 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}\right )\right )=\log \left (\frac {1}{81} \left (\frac {6^{2/3} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )^2}{A^{2/3} y(x)^2}+\frac {9 \sqrt [3]{6} \left (9 A-2 x^{3/2}+3 \sqrt {x} y(x)\right )}{\sqrt [3]{A} y(x)}+81\right )\right )+2 \log \left (3 \sqrt [3]{A}-\sqrt [3]{6} \sqrt {x}\right )+6 c_1,y(x)\right ] \]