Internal problem ID [10753]
Internal file name [OUTPUT/9701_Monday_June_06_2022_03_29_16_PM_5596839/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-\frac {3 y}{\left (a x +b \right )^{\frac {1}{3}} x^{\frac {5}{3}}}=\frac {3}{\left (a x +b \right )^{\frac {2}{3}} x^{\frac {7}{3}}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime } \left (a x +b \right )^{\frac {2}{3}} x^{\frac {7}{3}}-3 y \left (a x +b \right )^{\frac {1}{3}} x^{\frac {2}{3}}-3=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {3 y \left (a x +b \right )^{\frac {1}{3}} x^{\frac {2}{3}}+3}{y \left (a x +b \right )^{\frac {2}{3}} x^{\frac {7}{3}}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \left (x \right ) = -\frac {6 \sqrt {3}}{\left (a x +b \right )^{\frac {1}{3}} x^{\frac {2}{3}} \left (\left (\left (a x +b \right )^{\frac {1}{3}} x^{\frac {5}{3}} \left (\frac {a}{\left (a x +b \right )^{2} x^{4}}\right )^{\frac {1}{3}}+2\right ) \sqrt {3}+3 x^{\frac {5}{3}} \left (a x +b \right )^{\frac {1}{3}} \left (\frac {a}{\left (a x +b \right )^{2} x^{4}}\right )^{\frac {1}{3}} \tan \left (\operatorname {RootOf}\left (\sqrt {3}\, \ln \left (\frac {\tan \left (\textit {\_Z} \right )^{2}+1}{\left (\sqrt {3}+\tan \left (\textit {\_Z} \right )\right )^{2}}\right )+6 \sqrt {3}\, c_{1} -2 \sqrt {3}\, \left (\int \left (\frac {a}{\left (a x +b \right )^{2} x^{4}}\right )^{\frac {2}{3}} \left (a x +b \right )^{\frac {2}{3}} x^{\frac {7}{3}}d x \right )-6 \textit {\_Z} \right )\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.769 (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]
\[ \text {Solve}\left [\frac {1}{6} \left (2 \sqrt {3} \arctan \left (\frac {-\frac {2 \left (x^{2/3} y(x) \sqrt [3]{a x+b}+3\right )}{\sqrt [3]{a x^3} y(x)}-1}{\sqrt {3}}\right )+2 \log \left (\frac {-x^{2/3} y(x) \sqrt [3]{a x+b}-3}{\sqrt [3]{a x^3} y(x)}+1\right )-\log \left (\frac {\left (x^{2/3} y(x) \sqrt [3]{a x+b}+3\right )^2}{\left (a x^3\right )^{2/3} y(x)^2}+\frac {x^{2/3} y(x) \sqrt [3]{a x+b}+3}{\sqrt [3]{a x^3} y(x)}+1\right )\right )=\frac {\sqrt [3]{a} x \left (-\log \left (a^{2/3} x^{2/3}+\sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a x+b}+(a x+b)^{2/3}\right )+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{x}}{2 \sqrt [3]{a x+b}+\sqrt [3]{a} \sqrt [3]{x}}\right )+2 \log \left (\sqrt [3]{a x+b}-\sqrt [3]{a} \sqrt [3]{x}\right )\right )}{6 \sqrt [3]{a x^3}}+c_1,y(x)\right ] \]