Internal problem ID [10666]
Internal file name [OUTPUT/9614_Monday_June_06_2022_03_15_50_PM_41390442/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: 18.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-y=\frac {2 a^{2}}{\sqrt {8 a^{2}+x^{2}}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime } \sqrt {8 a^{2}+x^{2}}-y \sqrt {8 a^{2}+x^{2}}-2 a^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y \sqrt {8 a^{2}+x^{2}}+2 a^{2}}{y \sqrt {8 a^{2}+x^{2}}} \end {array} \]
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 <- Abel successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 715
dsolve(y(x)*diff(y(x),x)-y(x)=2*a^2/sqrt(x^2+8*a^2),y(x), singsol=all)
\[ \frac {512 \left (-\frac {33 \left (a^{4}+\frac {23}{66} a^{2} x^{2}+\frac {1}{66} x^{4}\right ) x \sqrt {8 a^{2}+x^{2}}}{64}+a^{6}+\frac {75 a^{4} x^{2}}{64}+\frac {27 a^{2} x^{4}}{128}+\frac {x^{6}}{128}\right ) {\mathrm e}^{-\frac {\left (-y \left (x \right )+x \right )^{2} \left (-64 \sqrt {8 a^{2}+x^{2}}\, a^{6}-108 \sqrt {8 a^{2}+x^{2}}\, a^{4} x^{2}-25 \sqrt {8 a^{2}+x^{2}}\, a^{2} x^{4}-\sqrt {8 a^{2}+x^{2}}\, x^{6}+328 a^{6} x +200 a^{4} x^{3}+29 a^{2} x^{5}+x^{7}\right )^{2}}{2 \left (128 a^{6}+150 a^{4} x^{2}-66 \sqrt {8 a^{2}+x^{2}}\, a^{4} x +27 a^{2} x^{4}-23 \sqrt {8 a^{2}+x^{2}}\, a^{2} x^{3}+x^{6}-\sqrt {8 a^{2}+x^{2}}\, x^{5}\right )^{2} a^{2} \left (-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}\right )}} a -128 \left (\sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {\left (-y \left (x \right )+x \right ) \sqrt {2}\, \left (-\left (-64 a^{6}-108 a^{4} x^{2}-25 a^{2} x^{4}-x^{6}\right ) \sqrt {8 a^{2}+x^{2}}-328 a^{6} x -200 a^{4} x^{3}-29 a^{2} x^{5}-x^{7}\right )}{2 \sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}\, \left (\left (-66 x \,a^{5}-23 a^{3} x^{3}-a \,x^{5}\right ) \sqrt {8 a^{2}+x^{2}}+128 a^{7}+150 a^{5} x^{2}+27 a^{3} x^{4}+a \,x^{6}\right )}\right )-c_{1} \right ) \left (\frac {\left (\left (a^{4}+\frac {21}{32} a^{2} x^{2}+\frac {1}{32} x^{4}\right ) y \left (x \right )-\frac {33 \left (a^{4}+\frac {23}{66} a^{2} x^{2}+\frac {1}{66} x^{4}\right ) x}{16}\right ) \sqrt {8 a^{2}+x^{2}}}{4}+\frac {\left (-25 a^{4} x -\frac {25}{4} a^{2} x^{3}-\frac {1}{4} x^{5}\right ) y \left (x \right )}{32}+a^{6}+\frac {75 a^{4} x^{2}}{64}+\frac {27 a^{2} x^{4}}{128}+\frac {x^{6}}{128}\right ) \sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}}{\sqrt {-\sqrt {8 a^{2}+x^{2}}\, x +4 a^{2}+x^{2}}\, \left (\left (\left (32 a^{4}+21 a^{2} x^{2}+x^{4}\right ) y \left (x \right )-66 a^{4} x -23 a^{2} x^{3}-x^{5}\right ) \sqrt {8 a^{2}+x^{2}}+\left (-100 a^{4} x -25 a^{2} x^{3}-x^{5}\right ) y \left (x \right )+128 a^{6}+150 a^{4} x^{2}+27 a^{2} x^{4}+x^{6}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-y[x]==2*a^2/Sqrt[x^2+8*a^2],y[x],x,IncludeSingularSolutions -> True]
Not solved