Internal problem ID [9909]
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: 3.
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 {2 x}{9}-A -\frac {B}{\sqrt {x}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 89
dsolve(y(x)*diff(y(x),x)-y(x)=-2/9*x+A+B*x^(-1/2),y(x), singsol=all)
\[ y \relax (x ) = -\frac {A \left (9 A \sqrt {x}-2 x^{\frac {3}{2}}+9 B \right )}{3 \left (A \sqrt {x}+\RootOf \left (9 A^{3} \left (\int _{}^{\textit {\_Z}}\frac {1}{-2 \textit {\_a}^{3} B^{2}+9 \textit {\_a} \,A^{3}-9 A^{3}}d \textit {\_a} \right )+\int -\frac {9 A}{2 \left (9 x A -2 x^{2}+9 B \sqrt {x}\right )}d x +c_{1}\right ) B \right )} \]
✓ Solution by Mathematica
Time used: 6.078 (sec). Leaf size: 415
DSolve[y[x]*y'[x]-y[x]==-2/9*x+A+B*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [6 \text {RootSum}\left [8 \text {$\#$1}^6-72 \text {$\#$1}^4 A-36 \text {$\#$1}^4 y(x)-72 \text {$\#$1}^3 B+162 \text {$\#$1}^2 A^2+162 \text {$\#$1}^2 A y(x)+54 \text {$\#$1}^2 y(x)^2+324 \text {$\#$1} A B+162 \text {$\#$1} B y(x)-81 A y(x)^2+162 B^2-27 y(x)^3\&,\frac {-2 \text {$\#$1}^3 \log \left (\sqrt {x}-\text {$\#$1}\right )+9 \text {$\#$1} A \log \left (\sqrt {x}-\text {$\#$1}\right )+9 B \log \left (\sqrt {x}-\text {$\#$1}\right )+9 \text {$\#$1} y(x) \log \left (\sqrt {x}-\text {$\#$1}\right )}{8 \text {$\#$1}^5-48 \text {$\#$1}^3 A-24 \text {$\#$1}^3 y(x)-36 \text {$\#$1}^2 B+54 \text {$\#$1} A^2+54 \text {$\#$1} A y(x)+18 \text {$\#$1} y(x)^2+54 A B+27 B y(x)}\&\right ]+\int _1^{y(x)}\left (\frac {162 K[1]}{8 x^3-72 A x^2-36 K[1] x^2-72 B x^{3/2}+162 A^2 x+54 K[1]^2 x+162 A K[1] x+324 A B \sqrt {x}+162 B K[1] \sqrt {x}-27 K[1]^3+162 B^2-81 A K[1]^2}+\frac {162 K[1]}{-8 x^3+72 A x^2+36 K[1] x^2+72 B x^{3/2}-162 A^2 x-54 K[1]^2 x-162 A K[1] x-324 A B \sqrt {x}-162 B K[1] \sqrt {x}+27 K[1]^3-162 B^2+81 A K[1]^2}\right )dK[1]=c_1,y(x)\right ] \]