Internal problem ID [10697]
Internal file name [OUTPUT/9645_Monday_June_06_2022_03_17_32_PM_34499080/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: 49.
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.
\[ \boxed {y y^{\prime }-y=2 x +2 A \left (10 \sqrt {x}+31 A +\frac {30 A^{2}}{\sqrt {x}}\right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 60 A^{3}+62 A^{2} \sqrt {x}+20 A x -y y^{\prime } \sqrt {x}+y \sqrt {x}+2 x^{\frac {3}{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-60 A^{3}-62 A^{2} \sqrt {x}-20 A x -y \sqrt {x}-2 x^{\frac {3}{2}}}{y \sqrt {x}} \end {array} \]
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: 0.016 (sec). Leaf size: 196
dsolve(y(x)*diff(y(x),x)-y(x)=2*x+2*A*(10*x^(1/2)+31*A+30*A^2*x^(-1/2)),y(x), singsol=all)
\[ c_{1} -\frac {\left (3 A +\sqrt {x}\right ) 2^{\frac {1}{3}} \left (\frac {12 A^{2}+10 A \sqrt {x}+2 x -y \left (x \right )}{6 A^{2}+2 A \sqrt {x}+y \left (x \right )}\right )^{\frac {1}{3}} \left (\frac {15 A^{2}+8 A \sqrt {x}+x +y \left (x \right )}{6 A^{2}+2 A \sqrt {x}+y \left (x \right )}\right )^{\frac {1}{6}} y \left (x \right )}{4 \sqrt {\frac {\left (3 A +\sqrt {x}\right )^{2}}{6 A^{2}+2 A \sqrt {x}+y \left (x \right )}}\, \left (6 A^{2}+2 A \sqrt {x}+y \left (x \right )\right ) A}-\left (\int _{}^{\frac {6 A \sqrt {x}+2 x -3 y \left (x \right )}{12 A^{2}+4 A \sqrt {x}+2 y \left (x \right )}}\frac {\left (\textit {\_a} +1\right )^{\frac {1}{3}} \left (2 \textit {\_a} +5\right )^{\frac {1}{6}}}{\sqrt {2 \textit {\_a} +3}}d \textit {\_a} \right ) = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==2*x+2*A*(10*x^(1/2)+31*A+30*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved