61.22.3 problem 3

Internal problem ID [12328]
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 1.3.1-2. Solvable equations and their solutions
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 07:54:23 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime } y-y&=-\frac {2 x}{9}+A +\frac {B}{\sqrt {x}} \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 90

dsolve(y(x)*diff(y(x),x)-y(x)=-2/9*x+A+B*x^(-1/2),y(x), singsol=all)
 

\[ y = -\frac {9 \left (A \sqrt {x}+B -\frac {2 x^{{3}/{2}}}{9}\right ) A}{3 A \sqrt {x}+3 \operatorname {RootOf}\left (18 A^{3} \left (\int _{}^{\textit {\_Z}}\frac {1}{-2 \textit {\_a}^{3} B^{2}+9 \textit {\_a} \,A^{3}-9 A^{3}}d \textit {\_a} \right )-9 A \left (\int \frac {1}{9 x A -2 x^{2}+9 B \sqrt {x}}d x \right )+2 c_{1} \right ) B} \]

✓ Solution by Mathematica

Time used: 0.323 (sec). Leaf size: 141

DSolve[y[x]*D[y[x],x]-y[x]==-2/9*x+A+B*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\frac {3 \sqrt [3]{3} B^{2/3}}{2^{2/3} \sqrt {K[2]} \left (\sqrt {K[2]} (2 K[2]-9 A)-9 B\right )}dK[2]+c_1=\int _1^{-\frac {\sqrt [3]{2} \left (9 B+3 \sqrt {x} y(x)+(9 A-2 x) \sqrt {x}\right )}{3\ 3^{2/3} \sqrt [3]{B} y(x)}}\frac {1}{K[1]^3-\frac {3^{2/3} A K[1]}{\sqrt [3]{2} B^{2/3}}+1}dK[1],y(x)\right ] \]