Internal problem ID [2012]
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.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {-\frac {\left (y^{\prime }\right )^{2}}{y^{2}}+\frac {y^{\prime \prime }}{y}+\frac {2 a \coth \left (2 a x \right ) y^{\prime }}{y}-2 a^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 29.656 (sec). Leaf size: 204
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)
\[ y \relax (x ) = {\mathrm e}^{\int \left (\int \frac {2 a^{2}}{\sqrt {\frac {\sinh \left (3 a x \right )-\sinh \left (a x \right )+\cosh \left (3 a x \right )-\cosh \left (a x \right )}{\cosh \left (3 a x \right )-\cosh \left (a x \right )}}\, \sqrt {\frac {-\cosh \left (3 a x \right )+\cosh \left (a x \right )+\sinh \left (3 a x \right )-\sinh \left (a x \right )}{\cosh \left (3 a x \right )-\cosh \left (a x \right )}}}d x \right ) \sqrt {\frac {\cosh \left (2 a x \right )+\sinh \left (2 a x \right )}{\sinh \left (2 a x \right )}}\, \sqrt {\frac {\cosh \left (2 a x \right )-\sinh \left (2 a x \right )}{\sinh \left (2 a x \right )}}d x} {\mathrm e}^{c_{1} \left (\int \sqrt {\frac {\cosh \left (a x \right )}{\sinh \left (a x \right )}+1-\frac {1}{2 \sinh \left (a x \right ) \cosh \left (a x \right )}}\, \sqrt {\frac {\cosh \left (a x \right )}{\sinh \left (a x \right )}-\frac {1}{2 \sinh \left (a x \right ) \cosh \left (a x \right )}-1}d x \right )} c_{2} \]
✓ Solution by Mathematica
Time used: 0.343 (sec). Leaf size: 258
DSolve[D[1/y[x]*y'[x],x]+(2*a*Coth[1/y[x]*y'[x]])==2*a^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \exp \left (\frac {\text {PolyLog}\left (2,\frac {(a-1) \exp \left (2 \text {InverseFunction}\left [\frac {\text {$\#$1} a+\log (a \sinh (\text {$\#$1})-\cosh (\text {$\#$1}))}{a^2-1}\&\right ][2 a x+c_1]\right )}{a+1}\right )+2 \text {InverseFunction}\left [\frac {\text {$\#$1} a+\log (a \sinh (\text {$\#$1})-\cosh (\text {$\#$1}))}{a^2-1}\&\right ][2 a x+c_1] \left (\log \left (1-\sqrt {\frac {a-1}{a+1}} \exp \left (\text {InverseFunction}\left [\frac {\text {$\#$1} a+\log (a \sinh (\text {$\#$1})-\cosh (\text {$\#$1}))}{a^2-1}\&\right ][2 a x+c_1]\right )\right )+\log \left (1+\sqrt {\frac {a-1}{a+1}} \exp \left (\text {InverseFunction}\left [\frac {\text {$\#$1} a+\log (a \sinh (\text {$\#$1})-\cosh (\text {$\#$1}))}{a^2-1}\&\right ][2 a x+c_1]\right )\right )\right )+(a-1) \text {InverseFunction}\left [\frac {\text {$\#$1} a+\log (a \sinh (\text {$\#$1})-\cosh (\text {$\#$1}))}{a^2-1}\&\right ][2 a x+c_1]{}^2}{4 a \left (a^2-1\right )}\right ) \\ \end{align*}