Internal problem ID [2778]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 1
Problem number: 23.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_linear]
Solve \begin {gather*} \boxed {y^{\prime }-4 \csc \relax (x ) x \left (1-\left (\tan ^{2}\relax (x )\right )+y\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.017 (sec). Leaf size: 178
dsolve(diff(y(x),x) = 4*csc(x)*x*(1-tan(x)^2+y(x)),y(x), singsol=all)
\[ y \relax (x ) = \left (1+{\mathrm e}^{i x}\right )^{-4 x} \left (1-{\mathrm e}^{i x}\right )^{4 x} {\mathrm e}^{-4 i \left (-\dilog \left (1+{\mathrm e}^{i x}\right )+\dilog \left (1-{\mathrm e}^{i x}\right )\right )} c_{1}+4 \left (1+{\mathrm e}^{i x}\right )^{-4 x} \left (1-{\mathrm e}^{i x}\right )^{4 x} {\mathrm e}^{-4 i \left (-\dilog \left (1+{\mathrm e}^{i x}\right )+\dilog \left (1-{\mathrm e}^{i x}\right )\right )} \left (\int \frac {4 \left (1+{\mathrm e}^{i x}\right )^{4 x} {\mathrm e}^{4 i \left (-\dilog \left (1+{\mathrm e}^{i x}\right )+\dilog \left (1-{\mathrm e}^{i x}\right )\right )} \left (1-{\mathrm e}^{i x}\right )^{-4 x} x \left (-\sin \left (3 x \right )+\sin \relax (x )\right )}{-1+\cos \left (4 x \right )}d x \right ) \]
✓ Solution by Mathematica
Time used: 1.312 (sec). Leaf size: 117
DSolve[y'[x]==2*Csc[x]*2*x*(1-Tan[x]^2+y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \exp \left (-8 i \text {PolyLog}\left (2,e^{i x}\right )+2 i \text {PolyLog}\left (2,e^{2 i x}\right )-8 x \tanh ^{-1}\left (e^{i x}\right )\right ) \left (\int _1^x4 \exp \left (8 \tanh ^{-1}\left (e^{i K[1]}\right ) K[1]+8 i \text {PolyLog}\left (2,e^{i K[1]}\right )-2 i \text {PolyLog}\left (2,e^{2 i K[1]}\right )\right ) \cos (2 K[1]) \csc (K[1]) K[1] \sec ^2(K[1])dK[1]+c_1\right ) \\ \end{align*}