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]]
\[ \boxed {-\frac {{y^{\prime }}^{2}}{y^{2}}+\frac {y^{\prime \prime }}{y}+\frac {2 a \coth \left (2 a x \right ) y^{\prime }}{y}-2 a^{2}=0} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 407
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)
\begin{align*} y \left (x \right ) = -\frac {2^{\frac {3}{4}} {\mathrm e}^{-a x} {\mathrm e}^{-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left ({\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{-2 a x}}{2}\right )}{c_{1} a}} {\mathrm e}^{-\frac {c_{2}}{c_{1}}} {\left (\left ({\mathrm e}^{8 a x}-2 \,{\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left ({\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{-2 a x}}{2}\right )}{c_{1} a}}\right )}^{\frac {1}{4}}}{2} \\ y \left (x \right ) = \frac {2^{\frac {3}{4}} {\mathrm e}^{-a x} {\mathrm e}^{-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left ({\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{-2 a x}}{2}\right )}{c_{1} a}} {\mathrm e}^{-\frac {c_{2}}{c_{1}}} {\left (\left ({\mathrm e}^{8 a x}-2 \,{\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left ({\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{-2 a x}}{2}\right )}{c_{1} a}}\right )}^{\frac {1}{4}}}{2} \\ y \left (x \right ) = -\frac {i 2^{\frac {3}{4}} {\mathrm e}^{-a x} {\mathrm e}^{-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left ({\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{-2 a x}}{2}\right )}{c_{1} a}} {\mathrm e}^{-\frac {c_{2}}{c_{1}}} {\left (\left ({\mathrm e}^{8 a x}-2 \,{\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left ({\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{-2 a x}}{2}\right )}{c_{1} a}}\right )}^{\frac {1}{4}}}{2} \\ y \left (x \right ) = \frac {i 2^{\frac {3}{4}} {\mathrm e}^{-a x} {\mathrm e}^{-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left ({\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{-2 a x}}{2}\right )}{c_{1} a}} {\mathrm e}^{-\frac {c_{2}}{c_{1}}} {\left (\left ({\mathrm e}^{8 a x}-2 \,{\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left ({\mathrm e}^{4 a x}+1\right ) {\mathrm e}^{-2 a x}}{2}\right )}{c_{1} a}}\right )}^{\frac {1}{4}}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.487 (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]
\begin{align*} 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 ) \\ \end{align*}