61.24.17 problem 17

Internal problem ID [12430]
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 1.3.3-2.
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 07:58:43 PM
CAS classification : [[_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime } y&=\frac {3 y}{\left (a x +b \right )^{{1}/{3}} x^{{5}/{3}}}+\frac {3}{\left (a x +b \right )^{{2}/{3}} x^{{7}/{3}}} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 143

dsolve(y(x)*diff(y(x),x)=3*(a*x+b)^(-1/3)*x^(-5/3)*y(x)+3*(a*x+b)^(-2/3)*x^(-7/3),y(x), singsol=all)
 

\[ y = -\frac {6 \sqrt {3}}{\left (a x +b \right )^{{1}/{3}} x^{{2}/{3}} \left (\left (\left (a x +b \right )^{{1}/{3}} x^{{5}/{3}} \left (\frac {a}{\left (a x +b \right )^{2} x^{4}}\right )^{{1}/{3}}+2\right ) \sqrt {3}+3 x^{{5}/{3}} \tan \left (\operatorname {RootOf}\left (\sqrt {3}\, \ln \left (\frac {\tan \left (\textit {\_Z} \right )^{2}+1}{\left (\tan \left (\textit {\_Z} \right )+\sqrt {3}\right )^{2}}\right )+6 \sqrt {3}\, c_{1} -2 \sqrt {3}\, \left (\int \left (\frac {a}{\left (a x +b \right )^{2} x^{4}}\right )^{{2}/{3}} \left (a x +b \right )^{{2}/{3}} x^{{7}/{3}}d x \right )-6 \textit {\_Z} \right )\right ) \left (\frac {a}{\left (a x +b \right )^{2} x^{4}}\right )^{{1}/{3}} \left (a x +b \right )^{{1}/{3}}\right )} \]

✓ Solution by Mathematica

Time used: 3.928 (sec). Leaf size: 111

DSolve[y[x]*D[y[x],x]==3*(a*x+b)^(-1/3)*x^(-5/3)*y[x]+3*(a*x+b)^(-2/3)*x^(-7/3),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {-x^{2/3} \sqrt [3]{b+a x} y(x)-3}{\sqrt [3]{a x^3} y(x)}}\frac {1}{K[1]^3+1}dK[1]+\frac {\left (a x^3\right )^{2/3} \left (\frac {a x}{b}+1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {2}{3},\frac {5}{3},-\frac {a x}{b}\right )}{2 x^{4/3} (a x+b)^{2/3}}=c_1,y(x)\right ] \]