Internal problem ID [10713]
Internal file name [OUTPUT/9661_Monday_June_06_2022_03_20_32_PM_46669422/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. Form \(y y'-y=f(x)\). subsection 1.3.1-2.
Solvable equations and their solutions
Problem number: 65.
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.
\[ \boxed {y y^{\prime }-y=-\frac {6 x}{25}+\frac {4 B^{2} \left (\left (2-A \right ) x^{\frac {1}{3}}-\frac {3 B \left (2 A +1\right )}{2}+\frac {B^{2} \left (1-3 A \right )}{x^{\frac {1}{3}}}-\frac {A \,B^{3}}{x^{\frac {2}{3}}}\right )}{75}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 4 A \,B^{5}+12 A \,B^{4} x^{\frac {1}{3}}+12 A \,B^{3} x^{\frac {2}{3}}+4 A x \,B^{2}-4 B^{4} x^{\frac {1}{3}}+6 B^{3} x^{\frac {2}{3}}-8 B^{2} x +75 y y^{\prime } x^{\frac {2}{3}}-75 y x^{\frac {2}{3}}+18 x^{\frac {5}{3}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-4 A \,B^{5}-12 A \,B^{4} x^{\frac {1}{3}}-12 A \,B^{3} x^{\frac {2}{3}}+4 B^{4} x^{\frac {1}{3}}-6 B^{3} x^{\frac {2}{3}}-18 x^{\frac {5}{3}}-4 A x \,B^{2}+8 B^{2} x +75 y x^{\frac {2}{3}}}{75 y x^{\frac {2}{3}}} \end {array} \]
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 Looking for potential symmetries Looking for potential symmetries <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 3187
dsolve(y(x)*diff(y(x),x)-y(x)=-6/25*x+4/75*B^2*((2-A)*x^(1/3)-3/2*B*(2*A+1)+B^2*(1-3*A)*x^(-1/3)-A*B^3*x^(-2/3)),y(x), singsol=all)
\[ \text {Expression too large to display} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-6/25*x+4/75*B^2*((2-A)*x^(1/3)-3/2*B*(2*A+1)+B^2*(1-3*A)*x^(-1/3)-A*B^3*x^(-2/3)),y[x],x,IncludeSingularSolutions -> True]
Not solved