Internal problem ID [10682]
Internal file name [OUTPUT/9630_Monday_June_06_2022_03_16_47_PM_24109604/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: 34.
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.
\[ \boxed {y y^{\prime }-y=A \left (n +2\right ) \left (\sqrt {x}+2 \left (n +2\right ) A +\frac {\left (n +1\right ) \left (n +3\right ) A^{2}}{\sqrt {x}}\right )} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & A^{3} n^{3}+6 A^{3} n^{2}+2 A^{2} \sqrt {x}\, n^{2}+11 A^{3} n +8 A^{2} \sqrt {x}\, n +A x n +6 A^{3}+8 A^{2} \sqrt {x}+2 A x -y y^{\prime } \sqrt {x}+y \sqrt {x}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {-A^{3} n^{3}-6 A^{3} n^{2}-2 A^{2} \sqrt {x}\, n^{2}-11 A^{3} n -8 A^{2} \sqrt {x}\, n -A x n -6 A^{3}-8 A^{2} \sqrt {x}-2 A x -y \sqrt {x}}{y \sqrt {x}} \end {array} \]
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: 0.016 (sec). Leaf size: 410
dsolve(y(x)*diff(y(x),x)-y(x)=A*(n+2)*(x^(1/2)+2*(n+2)*A+(n+1)*(n+3)*A^2*x^(-1/2)),y(x), singsol=all)
\[ \frac {\left (n +2\right ) \left (\operatorname {BesselI}\left (\frac {3+n}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right ) c_{1} +\operatorname {BesselK}\left (\frac {3+n}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right )\right ) A \sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}+\left (c_{1} \operatorname {BesselI}\left (\frac {1}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right )-\operatorname {BesselK}\left (\frac {1}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right )\right ) \left (\sqrt {x}+\left (n +1\right ) A \right )}{A \sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\, \left (n +2\right ) \operatorname {BesselI}\left (\frac {3+n}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right )+\left (\sqrt {x}+\left (n +1\right ) A \right ) \operatorname {BesselI}\left (\frac {1}{n +2}, -\sqrt {\frac {2 \left (n +2\right ) A \sqrt {x}+A^{2} \left (n^{2}+4 n +3\right )+x -y \left (x \right )}{\left (n +2\right )^{2} A^{2}}}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==A*(n+2)*(x^(1/2)+2*(n+2)*A+(n+1)*(n+3)*A^2*x^(-1/2)),y[x],x,IncludeSingularSolutions -> True]
Not solved