Internal problem ID [10665]
Internal file name [OUTPUT/9613_Monday_June_06_2022_03_15_49_PM_99424666/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: 17.
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.
\[ \boxed {y y^{\prime }-y=-\frac {x}{4}+\frac {A \left (\sqrt {x}+5 A +\frac {3 A^{2}}{\sqrt {x}}\right )}{4}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 3 A^{3}+5 A^{2} \sqrt {x}+A x -4 y y^{\prime } \sqrt {x}+4 y \sqrt {x}-x^{\frac {3}{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-3 A^{3}-5 A^{2} \sqrt {x}-A x -4 y \sqrt {x}+x^{\frac {3}{2}}}{4 y \sqrt {x}} \end {array} \]
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 found: 2 potential symmetries. Proceeding with integration step <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 257
dsolve(y(x)*diff(y(x),x)-y(x)=-1/4*x+1/4*A*(x^(1/2)+5*A+3*A^2*x^(-1/2)),y(x), singsol=all)
\[ -\frac {3 \left (-4 \left (\left (\int _{}^{\frac {6 A \sqrt {x}-2 x +3 y \left (x \right )}{12 A^{2}-4 A \sqrt {x}+2 y \left (x \right )}}\frac {{\mathrm e}^{-\frac {2}{2 \textit {\_a} +1}} \sqrt {2 \textit {\_a} +1}}{\sqrt {2 \textit {\_a} -3}}d \textit {\_a} \right ) A +\frac {c_{1}}{2}\right ) \left (A^{2}-\frac {A \sqrt {x}}{3}+\frac {y \left (x \right )}{6}\right ) \sqrt {-\frac {\left (3 A -\sqrt {x}\right )^{2}}{6 A^{2}-2 A \sqrt {x}+y \left (x \right )}}+\frac {\sqrt {\frac {3 A^{2}+2 A \sqrt {x}-x +2 y \left (x \right )}{6 A^{2}-2 A \sqrt {x}+y \left (x \right )}}\, y \left (x \right ) {\mathrm e}^{\frac {-6 A^{2}+2 A \sqrt {x}-y \left (x \right )}{3 A^{2}+2 A \sqrt {x}-x +2 y \left (x \right )}} \left (3 A -\sqrt {x}\right )}{3}\right )}{\sqrt {-\frac {\left (3 A -\sqrt {x}\right )^{2}}{6 A^{2}-2 A \sqrt {x}+y \left (x \right )}}\, \left (6 A^{2}-2 A \sqrt {x}+y \left (x \right )\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-1/4*x+1/4*A*(x^(1/2)+5*A+3*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved