Internal problem ID [10670]
Internal file name [OUTPUT/9618_Monday_June_06_2022_03_16_06_PM_10022587/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: 22.
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.
\[ \boxed {y y^{\prime }-y=-\frac {4 x}{25}+\frac {A}{\sqrt {x}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -25 y y^{\prime } \sqrt {x}+25 y \sqrt {x}-4 x^{\frac {3}{2}}+25 A =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-25 y \sqrt {x}+4 x^{\frac {3}{2}}-25 A}{25 y \sqrt {x}} \end {array} \]
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: 270
dsolve(y(x)*diff(y(x),x)-y(x)=-4/25*x+A*x^(-1/2),y(x), singsol=all)
\[ \frac {625 \sqrt {A \,x^{\frac {3}{2}}}\, c_{1} \left (-\frac {A y \left (x \right )^{2} \sqrt {x}}{2}+\frac {16 x^{4}}{625}-\frac {16 x^{3} y \left (x \right )}{125}+\frac {6 y \left (x \right )^{2} x^{2}}{25}-\frac {x y \left (x \right )^{3}}{5}+\frac {y \left (x \right )^{4}}{16}+A^{2} x +\frac {4 A y \left (x \right ) x^{\frac {3}{2}}}{5}-\frac {8 A \,x^{\frac {5}{2}}}{25}\right ) \sqrt {\frac {A \sqrt {x}-\frac {4 \left (x -\frac {5 y \left (x \right )}{4}\right )^{2}}{25}}{\sqrt {x}\, A}}+\frac {625 \left (y \left (x \right )^{4}-10 A^{2} y \left (x \right )\right ) x^{\frac {3}{2}}}{2}+500 \left (-y \left (x \right )^{3}+4 A^{2}\right ) x^{\frac {5}{2}}+\frac {128 x^{\frac {11}{2}}}{5}+400 y \left (x \right )^{2} x^{\frac {7}{2}}-160 y \left (x \right ) x^{\frac {9}{2}}+\frac {625 \left (-\frac {y \left (x \right )^{5}}{2}+5 A^{2} y \left (x \right )^{2}\right ) \sqrt {x}}{4}-400 A \,x^{4}+1500 A y \left (x \right ) x^{3}-1875 A y \left (x \right )^{2} x^{2}+\frac {3125 A y \left (x \right )^{3} x}{4}-3125 A^{3} x}{\sqrt {A \,x^{\frac {3}{2}}}\, \sqrt {\frac {A \sqrt {x}-\frac {4 \left (x -\frac {5 y \left (x \right )}{4}\right )^{2}}{25}}{\sqrt {x}\, A}}\, \left (25 A \sqrt {x}-4 \left (x -\frac {5 y \left (x \right )}{4}\right )^{2}\right )^{2}} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-4/25*x+A*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
Not solved