61.22.26 problem 26

Internal problem ID [12351]
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 : 26
Date solved : Tuesday, January 28, 2025 at 07:56:09 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 134

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

\[ y = \frac {2 \,3^{{5}/{6}} 2^{{1}/{3}} \left (-2 x^{{3}/{2}}+9 A \right )}{\sqrt {x}\, \left (\left (27 \tan \left (\operatorname {RootOf}\left (18 \,3^{{5}/{6}} 2^{{1}/{3}} \left (\int \frac {\left (\frac {A}{x^{{3}/{2}}}\right )^{{2}/{3}} \sqrt {x}}{-2 x^{{3}/{2}}+9 A}d x \right )+\sqrt {3}\, \ln \left (-8 \sqrt {3}\, \sin \left (\textit {\_Z} \right ) \cos \left (\textit {\_Z} \right )^{3}-8 \cos \left (\textit {\_Z} \right )^{4}-4 \sqrt {3}\, \cos \left (\textit {\_Z} \right ) \sin \left (\textit {\_Z} \right )+16 \cos \left (\textit {\_Z} \right )^{2}+1\right )-12 \sqrt {3}\, c_{1} -12 \textit {\_Z} \right )\right )-9 \sqrt {3}\right ) \left (\frac {A}{x^{{3}/{2}}}\right )^{{1}/{3}}-6 \,2^{{1}/{3}} 3^{{5}/{6}}\right )} \]

✓ Solution by Mathematica

Time used: 0.254 (sec). Leaf size: 174

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

\[ \text {Solve}\left [6 \int _1^{-\frac {\sqrt [3]{2} \left (-2 x^{3/2}+3 y(x) \sqrt {x}+9 A\right )}{3\ 3^{2/3} \sqrt [3]{A} y(x)}}\frac {1}{K[1]^3+1}dK[1]+\log \left (3 \sqrt [3]{3} A^{2/3}+\sqrt [3]{2} 3^{2/3} \sqrt [3]{A} \sqrt {x}+2^{2/3} x\right )+2 \sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{6} \sqrt {x}}{\sqrt [3]{A}}+3}{3 \sqrt {3}}\right )=2 \log \left (3^{2/3} \sqrt [3]{A}-\sqrt [3]{2} \sqrt {x}\right )+6 c_1,y(x)\right ] \]