Internal problem ID [5632]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 2. Problems for Review and Discovery. Drill excercises. Page 105
Problem number: 2(g).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _quadrature]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\tan \relax (x )=0} \end {gather*} With initial conditions \begin {align*} [y \relax (1) = 1, y^{\prime }\relax (1) = -1] \end {align*}
✓ Solution by Maple
Time used: 0.594 (sec). Leaf size: 157
dsolve([diff(y(x),x$2)=tan(x),y(1) = 1, D(y)(1) = -1],y(x), singsol=all)
\[ y \relax (x ) = \frac {-i \left (1+{\mathrm e}^{2 i}\right ) \cos \relax (1) \polylog \left (2, -{\mathrm e}^{2 i x}\right )+2 x \cos \relax (1) \left (1+{\mathrm e}^{2 i}\right ) \ln \left (1+{\mathrm e}^{2 i x}\right )+i \left (1+{\mathrm e}^{2 i}\right ) \cos \relax (1) \polylog \left (2, -{\mathrm e}^{2 i}\right )-2 \cos \relax (1) \left (1+{\mathrm e}^{2 i}\right ) \ln \left (1+{\mathrm e}^{2 i}\right )+\left (\left (2 \ln \left (\cos \relax (1)\right ) x -2 x \ln \left (\cos \relax (x )\right )+4+3 i-i x^{2}+\left (-2-2 i\right ) x \right ) \cos \relax (1)-2 \sin \relax (1) \left (x -1\right )\right ) {\mathrm e}^{2 i}+\left (2 \ln \left (\cos \relax (1)\right ) x -2 x \ln \left (\cos \relax (x )\right )+4-i-i x^{2}+\left (-2+2 i\right ) x \right ) \cos \relax (1)-2 \sin \relax (1) \left (x -1\right )}{2 \cos \relax (1) \left (1+{\mathrm e}^{2 i}\right )} \]
✓ Solution by Mathematica
Time used: 0.046 (sec). Leaf size: 82
DSolve[{y''[x]==Tan[x],{y[1]==1,y'[1]==-1}},y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (-i \text {PolyLog}\left (2,-e^{2 i x}\right )+i \text {PolyLog}\left (2,-e^{2 i}\right )+(-2-i x) x+2 x \log \left (\left (1+e^{2 i x}\right ) \cos (1)\right )-2 x \log (\cos (x))+(4+i)-2 \log \left (1+e^{2 i}\right )\right ) \\ \end{align*}