Internal problem ID [10712]
Internal file name [OUTPUT/9660_Monday_June_06_2022_03_20_26_PM_79963363/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: 64.
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.
\[ \boxed {y y^{\prime }-y=-\frac {12 x}{49}+3 A \left (\frac {1}{49}+B \right ) \sqrt {x}+3 A^{2} \left (\frac {4}{49}-\frac {5 B}{2}\right )+\frac {15 A^{3} \left (\frac {1}{49}-\frac {5 B}{4}\right )}{4 \sqrt {x}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 3675 A^{3} B +5880 A^{2} \sqrt {x}\, B -2352 A B x -60 A^{3}-192 A^{2} \sqrt {x}-48 A x +784 y y^{\prime } \sqrt {x}-784 y \sqrt {x}+192 x^{\frac {3}{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-3675 A^{3} B -5880 A^{2} \sqrt {x}\, B +60 A^{3}+192 A^{2} \sqrt {x}+2352 A B x -192 x^{\frac {3}{2}}+48 A x +784 y \sqrt {x}}{784 y \sqrt {x}} \end {array} \]
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 Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 2 trying symmetry patterns for 1st order ODEs -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x)-(1/2)*y(x)*(1225*A^3*B+784*x*B*A-20*A^3+16*A*x-128*x^(3/2))/(1960*A^2*B*x^(3/2)+1 Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x)+(49*A*B*y(x)+A*y(x)+6*x)/(A*(1+49*B)*x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+(1/15)*(3675*A^3*B*y(x)-60*A^3*y(x)-2940*A^2*B*x+96*A^2*x+392*x^2)/(A^3*x*(-4+245 Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+3*A*(245*A*B*y(x)-8*A*y(x)+98*B*x+2*x)/(x*(735*A^2*B-24*A^2-98*x)), y(x)` ** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> trying a symmetry pattern of the form [F(x),G(x)] -> trying a symmetry pattern of the form [F(y),G(y)] -> trying a symmetry pattern of the form [F(x)+G(y), 0] -> trying a symmetry pattern of the form [0, F(x)+G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] -> trying a symmetry pattern of conformal type`
✗ Solution by Maple
dsolve(y(x)*diff(y(x),x)-y(x)=-12/49*x+3*A*(1/49+B)*x^(1/2)+3*A^2*(4/49-5/2*B)+15/4*A^3*(1/49-5/4*B)*x^(-1/2),y(x), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-12/49*x+3*A*(1/49+B)*x^(1/2)+3*A^2*(4/49-5/2*B)+15/4*A^3*(1/49-5/4*B)*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
Not solved