Internal problem ID [10788]
Internal file name [OUTPUT/9736_Monday_June_06_2022_04_48_31_PM_33126839/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: 52.
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.
\[ \boxed {y y^{\prime }-\frac {a \left (2 x -1\right ) y}{x^{\frac {5}{2}}}=\frac {a^{2} \left (x -1\right ) \left (3 x +1\right )}{2 x^{4}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 y y^{\prime } x^{\frac {13}{2}}-3 x^{\frac {9}{2}} a^{2}+2 x^{\frac {7}{2}} a^{2}-4 a y x^{5}+x^{\frac {5}{2}} a^{2}+2 a y x^{4}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {3 x^{\frac {9}{2}} a^{2}-2 x^{\frac {7}{2}} a^{2}+4 a y x^{5}-x^{\frac {5}{2}} a^{2}-2 a y x^{4}}{2 y x^{\frac {13}{2}}} \end {array} \]
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: 189
dsolve(y(x)*diff(y(x),x)-a*(2*x-1)*x^(-5/2)*y(x)=1/2*a^2*(x-1)*(3*x+1)*x^(-4),y(x), singsol=all)
\[ \frac {\frac {18 \sqrt {\frac {\left (-1+x \right ) a +y \left (x \right ) x^{\frac {3}{2}}}{x \left (y \left (x \right ) \sqrt {x}+a \right )}}\, \sqrt {5}\, 7^{\frac {5}{6}} \left (x +\frac {3}{2}\right ) \left (\frac {\left (-3 x -1\right ) a -3 y \left (x \right ) x^{\frac {3}{2}}}{x \left (y \left (x \right ) \sqrt {x}+a \right )}\right )^{\frac {1}{6}}}{1225}+1458 \left (\int _{}^{\frac {-\frac {18 y \left (x \right ) x^{\frac {3}{2}}}{35}+\frac {9 \left (-2 x -3\right ) a}{35}}{x \left (y \left (x \right ) \sqrt {x}+a \right )}}\frac {\textit {\_a} \left (5 \textit {\_a} -9\right )^{\frac {1}{6}} \sqrt {7 \textit {\_a} +9}}{\left (35 \textit {\_a} +18\right )^{\frac {2}{3}} \left (1225 \textit {\_a}^{3}-3159 \textit {\_a} -1458\right )}d \textit {\_a} +\frac {c_{1}}{1458}\right ) x \left (-\frac {a}{x \left (y \left (x \right ) \sqrt {x}+a \right )}\right )^{\frac {2}{3}}}{x \left (-\frac {a}{x \left (y \left (x \right ) \sqrt {x}+a \right )}\right )^{\frac {2}{3}}} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-a*(2*x-1)*x^(-5/2)*y[x]==1/2*a^2*(x-1)*(3*x+1)*x^(-4),y[x],x,IncludeSingularSolutions -> True]
Not solved