18.2.9 problem Problem 15.9(b)
Internal
problem
ID
[3492]
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)
Date
solved
:
Tuesday, January 28, 2025 at 02:39:16 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} -\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} \end{align*}
✓ Solution by Maple
Time used: 0.047 (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: 60.489 (sec). Leaf size: 287
DSolve[D[1/y[x]*D[y[x],x],x]+(2*a*Coth[1/y[x]*D[y[x],x]])==2*a^2,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to c_2 \exp \left (\frac {-\operatorname {PolyLog}\left (2,\frac {(a+1) \exp \left (-2 \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1]\right )}{a-1}\right )+2 \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1] \log \left (1-\frac {(a+1) \exp \left (-2 \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1]\right )}{a-1}\right )+(a+1) \text {InverseFunction}\left [\frac {-((a+1) \log (1-\tanh (\text {$\#$1})))+(a-1) \log (\tanh (\text {$\#$1})+1)+2 \log (1-a \tanh (\text {$\#$1}))}{2 \left (a^2-1\right )}\&\right ][2 a x+c_1]{}^2}{4 a \left (a^2-1\right )}\right )
\]