Internal problem ID [2521]
Internal file name [OUTPUT/2013_Sunday_June_05_2022_02_44_19_AM_88354591/index.tex]
Book: Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition,
2002
Section: Chapter 15, Higher order ordinary differential equations. 15.4 Exercises, page
523
Problem number: Problem 15.9(b).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {0=\frac {{y^{\prime }}^{2}}{y^{2}}-\frac {y^{\prime \prime }}{y}-\frac {2 a \coth \left (2 a x \right ) y^{\prime }}{y}+2 a^{2}} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful <- 2nd order, 2 integrating factors of the form mu(x,y) successful`
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 53
dsolve(diff( 1/y(x)*diff(y(x),x),x)+(2*a*coth(2*a*x))*(1/y(x)*diff(y(x),x))=2*a^2,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\frac {-x \,a^{2}+c_{1} \operatorname {arctanh}\left ({\mathrm e}^{2 a x}\right )-c_{2}}{a}} \sqrt {{\mathrm e}^{a x}-1}\, \sqrt {{\mathrm e}^{a x}+1}\, \sqrt {{\mathrm e}^{2 a x}+1} \]
✓ Solution by Mathematica
Time used: 60.504 (sec). Leaf size: 287
DSolve[D[1/y[x]*y'[x],x]+(2*a*Coth[1/y[x]*y'[x]])==2*a^2,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \exp \left (\frac {-\operatorname {PolyLog}\left (2,\frac {(a+1) \exp \left (-2 \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1]\right )}{a-1}\right )+2 \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1] \log \left (1-\frac {(a+1) \exp \left (-2 \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1]\right )}{a-1}\right )+(a+1) \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1]{}^2}{4 a \left (a^2-1\right )}\right ) \]