Internal problem ID [10706]
Internal file name [OUTPUT/9654_Monday_June_06_2022_03_17_57_PM_56089828/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: 58.
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.
\[ \boxed {y y^{\prime }-y=-\frac {12 x}{49}+\frac {2 A \left (\sqrt {x}+166 A +\frac {55 A^{2}}{\sqrt {x}}\right )}{49}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 110 A^{3}+332 A^{2} \sqrt {x}+2 A x -49 y y^{\prime } \sqrt {x}+49 y \sqrt {x}-12 x^{\frac {3}{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-110 A^{3}-332 A^{2} \sqrt {x}-2 A x -49 y \sqrt {x}+12 x^{\frac {3}{2}}}{49 y \sqrt {x}} \end {array} \]
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 <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 686
dsolve(y(x)*diff(y(x),x)-y(x)=-12/49*x+2/49*A*(x^(1/2)+166*A+55*A^2*x^(-1/2)),y(x), singsol=all)
\[ c_{1} +\frac {3 \sqrt {6}\, 4^{\frac {2}{3}} \left (\left (\sqrt {-35 A^{2}-7 A \sqrt {x}}\, \left (A \left (3+\frac {5 i \sqrt {6}}{3}\right ) \sqrt {x}+\frac {i \left (25 A^{2}+x \right ) \sqrt {6}}{6}-10 A^{2}+x -\frac {7 y \left (x \right )}{4}\right )-\frac {7 i x \sqrt {6}\, A}{6}+\left (-\frac {7 y \left (x \right )}{4}+5 A^{2}-\frac {35 i A^{2} \sqrt {6}}{3}\right ) \sqrt {x}-\frac {175 i A^{3} \sqrt {6}}{6}-50 A^{3}+\left (8 x -\frac {35 y \left (x \right )}{4}\right ) A +x^{\frac {3}{2}}\right ) \operatorname {hypergeom}\left (\left [-1, -\frac {1}{6}\right ], \left [\frac {2}{3}\right ], \frac {4 i \left (5 A +\sqrt {x}\right ) \sqrt {6}\, \sqrt {-35 A^{2}-7 A \sqrt {x}}}{10 i \sqrt {6}\, \left (A +\frac {\sqrt {x}}{5}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-120 A^{2}+36 A \sqrt {x}+12 x -21 y \left (x \right )}\right )+\frac {\sqrt {-35 A^{2}-7 A \sqrt {x}}\, \left (A \left (-7-5 i \sqrt {6}\right ) \sqrt {x}+\frac {i \left (-25 A^{2}-x \right ) \sqrt {6}}{2}-35 A^{2}\right )}{2}+\frac {175 \left (A +\frac {\sqrt {x}}{5}\right )^{2} A \left (i \sqrt {6}-2\right )}{4}\right )}{4 \left (\left (\left (2 \left (-18 i-7 \sqrt {6}\right ) A \sqrt {x}-70 A^{2} \sqrt {6}+120 i A^{2}-12 i x +21 i y \left (x \right )\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}+350 \left (-\frac {\sqrt {6}\, x}{25}+\frac {2 \left (-\sqrt {6}+\frac {9 i}{5}\right ) A \sqrt {x}}{5}-A^{2} \sqrt {6}-\frac {12 i A^{2}}{5}+\frac {6 i x}{25}-\frac {21 i y \left (x \right )}{50}\right ) A \right ) \operatorname {hypergeom}\left (\left [-\frac {2}{3}, \frac {1}{6}\right ], \left [\frac {4}{3}\right ], \frac {4 i \left (5 A +\sqrt {x}\right ) \sqrt {6}\, \sqrt {-35 A^{2}-7 A \sqrt {x}}}{10 i \sqrt {6}\, \left (A +\frac {\sqrt {x}}{5}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-120 A^{2}+36 A \sqrt {x}+12 x -21 y \left (x \right )}\right )+175 \left (\sqrt {-35 A^{2}-7 A \sqrt {x}}\, \left (\frac {A \sqrt {x}\, \left (\frac {6 i}{7}+\frac {\sqrt {6}}{5}\right )}{5}+\frac {A^{2} \sqrt {6}}{5}+\frac {3 i A^{2}}{7}+\frac {3 i x}{175}\right )+\left (\sqrt {6}-3 i\right ) \left (A +\frac {\sqrt {x}}{5}\right )^{2} A \right ) \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {7}{6}\right ], \left [\frac {7}{3}\right ], \frac {4 i \left (5 A +\sqrt {x}\right ) \sqrt {6}\, \sqrt {-35 A^{2}-7 A \sqrt {x}}}{10 i \sqrt {6}\, \left (A +\frac {\sqrt {x}}{5}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-120 A^{2}+36 A \sqrt {x}+12 x -21 y \left (x \right )}\right )\right ) {\left (\frac {i \left (5 A +\sqrt {x}\right ) \sqrt {6}\, \sqrt {-35 A^{2}-7 A \sqrt {x}}}{10 i \sqrt {6}\, \left (A +\frac {\sqrt {x}}{5}\right ) \sqrt {-35 A^{2}-7 A \sqrt {x}}-120 A^{2}+36 A \sqrt {x}+12 x -21 y \left (x \right )}\right )}^{\frac {1}{3}}} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-12/49*x+2/49*A*(x^(1/2)+166*A+55*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved