Internal problem ID [5631]
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]]
\[ \boxed {y^{\prime \prime }-\tan \left (x \right )=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = -1] \end {align*}
✓ Solution by Maple
Time used: 0.579 (sec). Leaf size: 135
dsolve([diff(y(x),x$2)=tan(x),y(1) = 1, D(y)(1) = -1],y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-i {\mathrm e}^{2 i}-i\right ) \operatorname {polylog}\left (2, -{\mathrm e}^{2 i x}\right )+2 x \left (1+{\mathrm e}^{2 i}\right ) \ln \left (1+{\mathrm e}^{2 i x}\right )+\left (i {\mathrm e}^{2 i}+i\right ) \operatorname {polylog}\left (2, -{\mathrm e}^{2 i}\right )+\left (-2 \,{\mathrm e}^{2 i}-2\right ) \ln \left (1+{\mathrm e}^{2 i}\right )+\left (2 \ln \left (\cos \left (1\right )\right ) x -2 x \ln \left (\cos \left (x \right )\right )+\left (-2 x +2\right ) \tan \left (1\right )-i x^{2}+\left (-2-2 i\right ) x +4+3 i\right ) {\mathrm e}^{2 i}+2 \ln \left (\cos \left (1\right )\right ) x -2 x \ln \left (\cos \left (x \right )\right )+\left (-2 x +2\right ) \tan \left (1\right )-i x^{2}+\left (-2+2 i\right ) x +4-i}{2 \,{\mathrm e}^{2 i}+2} \]
✓ Solution by Mathematica
Time used: 0.037 (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 \operatorname {PolyLog}\left (2,-e^{2 i x}\right )+i \operatorname {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*}