Internal problem ID [10767]
Internal file name [OUTPUT/9715_Monday_June_06_2022_04_47_30_PM_61373009/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: 31.
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.
\[ \boxed {y y^{\prime }-\frac {a \left (x +1\right ) y}{2 x^{\frac {7}{4}}}=\frac {a^{2} \left (x -1\right ) \left (3 x +5\right )}{4 x^{\frac {5}{2}}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 y y^{\prime } x^{\frac {17}{4}}-3 x^{\frac {15}{4}} a^{2}-2 x^{\frac {11}{4}} a^{2}+5 a^{2} x^{\frac {7}{4}}-2 a y x^{\frac {7}{2}}-2 x^{\frac {5}{2}} y a =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {3 x^{\frac {15}{4}} a^{2}+2 x^{\frac {11}{4}} a^{2}-5 a^{2} x^{\frac {7}{4}}+2 a y x^{\frac {7}{2}}+2 x^{\frac {5}{2}} y a}{4 y x^{\frac {17}{4}}} \end {array} \]
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)-1/2*a*(x+1)*x^(-7/4)*y(x)=1/4*a^2*(x-1)*(3*x+5)*x^(-5/2),y(x), singsol=all)
\[ \frac {\frac {36 \sqrt {-\frac {\left (-1+x \right ) a +x^{\frac {3}{4}} y \left (x \right )}{x^{\frac {3}{4}} \left (y \left (x \right )+x^{\frac {1}{4}} a \right )}}\, \sqrt {13}\, 55^{\frac {1}{6}} \left (x -\frac {15}{2}\right ) \left (\frac {\left (3 x +5\right ) a +3 x^{\frac {3}{4}} y \left (x \right )}{x^{\frac {3}{4}} \left (y \left (x \right )+x^{\frac {1}{4}} a \right )}\right )^{\frac {5}{6}}}{20449}+1458000 \left (\int _{}^{-\frac {90 \left (2 x^{\frac {3}{4}} y \left (x \right )+2 a x -15 a \right )}{143 \left (x^{\frac {3}{4}} y \left (x \right )+a x \right )}}\frac {\textit {\_a} \sqrt {11 \textit {\_a} -90}\, \left (13 \textit {\_a} +90\right )^{\frac {5}{6}}}{\left (143 \textit {\_a} +180\right )^{\frac {4}{3}} \left (20449 \textit {\_a}^{3}-1190700 \textit {\_a} -1458000\right )}d \textit {\_a} +\frac {c_{1}}{1458000}\right ) \left (\frac {a}{x^{\frac {3}{4}} \left (y \left (x \right )+x^{\frac {1}{4}} a \right )}\right )^{\frac {4}{3}} x}{\left (\frac {a}{x^{\frac {3}{4}} \left (y \left (x \right )+x^{\frac {1}{4}} a \right )}\right )^{\frac {4}{3}} x} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-1/2*a*(x+1)*x^(-7/4)*y[x]==1/4*a^2*(x-1)*(3*x+5)*x^(-5/2),y[x],x,IncludeSingularSolutions -> True]
Timed out