Internal problem ID [10672]
Internal file name [OUTPUT/9620_Monday_June_06_2022_03_16_08_PM_55707895/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: 24.
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.
\[ \boxed {y y^{\prime }-y=-\frac {12 x}{49}+\frac {2 A \left (5 \sqrt {x}+34 A +\frac {15 A^{2}}{\sqrt {x}}\right )}{49}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 30 A^{3}+68 A^{2} \sqrt {x}+10 A x -49 y y^{\prime } \sqrt {x}+49 y \sqrt {x}-12 x^{\frac {3}{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-30 A^{3}-68 A^{2} \sqrt {x}-10 A x -49 y \sqrt {x}+12 x^{\frac {3}{2}}}{49 y \sqrt {x}} \end {array} \]
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 found: 2 potential symmetries. Proceeding with integration step <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 270
dsolve(y(x)*diff(y(x),x)-y(x)=-12/49*x+2/49*A*(5*x^(1/2)+34*A+15*A^2*x^(-1/2)),y(x), singsol=all)
\[ \frac {\left (3 A -\sqrt {x}\right ) \left (36 A^{4}+120 A^{3} \sqrt {x}-80 A \,x^{\frac {3}{2}}+52 A^{2} x +84 A^{2} y \left (x \right )+140 A \sqrt {x}\, y \left (x \right )+16 x^{2}-56 y \left (x \right ) x +49 y \left (x \right )^{2}\right ) y \left (x \right )}{8 \sqrt {-\frac {\left (3 A -\sqrt {x}\right )^{2}}{6 A^{2}-2 A \sqrt {x}+y \left (x \right )}}\, \left (\frac {15 A^{2}+4 A \sqrt {x}-3 x +7 y \left (x \right )}{6 A^{2}-2 A \sqrt {x}+y \left (x \right )}\right )^{\frac {3}{2}} \left (6 A^{2}-2 A \sqrt {x}+y \left (x \right )\right )^{3} A}+\frac {\left (-54 A^{2}-6 A \sqrt {x}+8 x -21 y \left (x \right )\right ) \sqrt {-\frac {\left (3 A -\sqrt {x}\right )^{2}}{6 A^{2}-2 A \sqrt {x}+y \left (x \right )}}}{\sqrt {\frac {15 A^{2}+4 A \sqrt {x}-3 x +7 y \left (x \right )}{6 A^{2}-2 A \sqrt {x}+y \left (x \right )}}\, \left (36 A^{2}-12 A \sqrt {x}+6 y \left (x \right )\right )}+c_{1} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==-12/49*x+2/49*A*(5*x^(1/2)+34*A+15*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved