Internal problem ID [10730]
Internal file name [OUTPUT/9678_Monday_June_06_2022_03_21_43_PM_99671830/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 1.3.2. Equations of
the form \(y y'=f(x) y+1\)
Problem number: 6.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-\left (\frac {a}{x^{\frac {2}{3}}}-\frac {2}{3 a \,x^{\frac {1}{3}}}\right ) y=1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 3 y y^{\prime } x^{\frac {2}{3}} a -3 y a^{2}-3 x^{\frac {2}{3}} a +2 y x^{\frac {1}{3}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {3 x^{\frac {2}{3}} a +3 y a^{2}-2 y x^{\frac {1}{3}}}{3 x^{\frac {2}{3}} a y} \end {array} \]
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 Looking for potential symmetries Looking for potential symmetries <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 187
dsolve(y(x)*diff(y(x),x)=(a*x^(-2/3)-2/3*a^(-1)*x^(-1/3))*y(x)+1,y(x), singsol=all)
\[ \frac {-\sqrt {\frac {x^{\frac {2}{3}}+a y \left (x \right )}{a^{4}}}\, \operatorname {BesselI}\left (1, \frac {2 \sqrt {\frac {x^{\frac {2}{3}}+a y \left (x \right )}{a^{4}}}}{3}\right ) c_{1} a^{2}+\operatorname {BesselK}\left (1, -\frac {2 \sqrt {\frac {x^{\frac {2}{3}}+a y \left (x \right )}{a^{4}}}}{3}\right ) \sqrt {\frac {x^{\frac {2}{3}}+a y \left (x \right )}{a^{4}}}\, a^{2}+x^{\frac {1}{3}} \operatorname {BesselI}\left (0, \frac {2 \sqrt {\frac {x^{\frac {2}{3}}+a y \left (x \right )}{a^{4}}}}{3}\right ) c_{1} -x^{\frac {1}{3}} \operatorname {BesselK}\left (0, -\frac {2 \sqrt {\frac {x^{\frac {2}{3}}+a y \left (x \right )}{a^{4}}}}{3}\right )}{-\operatorname {BesselI}\left (1, \frac {2 \sqrt {\frac {x^{\frac {2}{3}}+a y \left (x \right )}{a^{4}}}}{3}\right ) \sqrt {\frac {x^{\frac {2}{3}}+a y \left (x \right )}{a^{4}}}\, a^{2}+x^{\frac {1}{3}} \operatorname {BesselI}\left (0, \frac {2 \sqrt {\frac {x^{\frac {2}{3}}+a y \left (x \right )}{a^{4}}}}{3}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==(a*x^(-2/3)-2/3*a^(-1)*x^(-1/3))*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
Not solved