Internal problem ID [10011]
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. Equations of
the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 17.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y y^{\prime }-\frac {3 y}{\left (a x +b \right )^{\frac {1}{3}} x^{\frac {5}{3}}}-\frac {3}{\left (a x +b \right )^{\frac {2}{3}} x^{\frac {7}{3}}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 147
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)
\[ y \relax (x ) = -\frac {6 \sqrt {3}}{-3 \tan \left (\RootOf \left (\sqrt {3}\, \ln \left (\frac {\tan ^{2}\left (\textit {\_Z} \right )+1}{\left (-\sqrt {3}+\tan \left (\textit {\_Z} \right )\right )^{2}}\right )+6 \sqrt {3}\, c_{1}-2 \sqrt {3}\, \left (\int \left (\frac {a}{\left (x a +b \right )^{2} x^{4}}\right )^{\frac {2}{3}} \left (x a +b \right )^{\frac {2}{3}} x^{\frac {7}{3}}d x \right )+6 \textit {\_Z} \right )\right ) \left (x a +b \right )^{\frac {2}{3}} x^{\frac {7}{3}} \left (\frac {a}{\left (x a +b \right )^{2} x^{4}}\right )^{\frac {1}{3}}+\left (x a +b \right )^{\frac {2}{3}} x^{\frac {7}{3}} \left (\frac {a}{\left (x a +b \right )^{2} x^{4}}\right )^{\frac {1}{3}} \sqrt {3}+2 \left (x a +b \right )^{\frac {1}{3}} x^{\frac {2}{3}} \sqrt {3}} \]
✓ Solution by Mathematica
Time used: 1.728 (sec). Leaf size: 312
DSolve[y[x]*y'[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]
\[ \text {Solve}\left [\frac {1}{6} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {-\frac {2 \left (x^{2/3} y(x) \sqrt [3]{a x+b}+3\right )}{\sqrt [3]{a x^3} y(x)}-1}{\sqrt {3}}\right )+2 \log \left (\frac {-x^{2/3} y(x) \sqrt [3]{a x+b}-3}{\sqrt [3]{a x^3} y(x)}+1\right )-\log \left (\frac {\left (x^{2/3} y(x) \sqrt [3]{a x+b}+3\right )^2}{\left (a x^3\right )^{2/3} y(x)^2}+\frac {x^{2/3} y(x) \sqrt [3]{a x+b}+3}{\sqrt [3]{a x^3} y(x)}+1\right )\right )=\frac {\sqrt [3]{a} x \left (-\log \left (a^{2/3} x^{2/3}+\sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a x+b}+(a x+b)^{2/3}\right )+2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{x}}{2 \sqrt [3]{a x+b}+\sqrt [3]{a} \sqrt [3]{x}}\right )+2 \log \left (\sqrt [3]{a x+b}-\sqrt [3]{a} \sqrt [3]{x}\right )\right )}{6 \sqrt [3]{a x^3}}+c_1,y(x)\right ] \]