Internal problem ID [10701]
Internal file name [OUTPUT/9649_Monday_June_06_2022_03_17_48_PM_70219559/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: 53.
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 A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-y=-\frac {12 x}{49}+A \sqrt {x}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-y=-\frac {12 x}{49}+A \sqrt {x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y-\frac {12 x}{49}+A \sqrt {x}}{y} \end {array} \]
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: 133
dsolve(y(x)*diff(y(x),x)-y(x)=-12/49*x+A*x^(1/2),y(x), singsol=all)
\[ \frac {196^{\frac {5}{6}} \left (\left (\frac {4 \left (x -\frac {7 y \left (x \right )}{4}\right ) \sqrt {3}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {7}{6}\right ], \left [\frac {3}{2}\right ], \frac {3 \left (-4 x +7 y \left (x \right )\right )^{2}}{196 x^{\frac {3}{2}} A}\right )}{7}+\sqrt {x}\, \sqrt {A \sqrt {x}}\, c_{1} \right ) 196^{\frac {1}{6}} \left (\frac {A \,x^{\frac {3}{2}}-\frac {12 \left (x -\frac {7 y \left (x \right )}{4}\right )^{2}}{49}}{x^{\frac {3}{2}} A}\right )^{\frac {1}{6}}-7 \,14^{\frac {1}{3}} A \sqrt {3}\, \sqrt {x}\right )}{196 \left (\frac {A \,x^{\frac {3}{2}}-\frac {12 \left (x -\frac {7 y \left (x \right )}{4}\right )^{2}}{49}}{x^{\frac {3}{2}} A}\right )^{\frac {1}{6}} \sqrt {A \sqrt {x}}\, \sqrt {x}} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-12/49*x+A*x^(1/2),y[x],x,IncludeSingularSolutions -> True]
Not solved