29.1.24 problem 23

Internal problem ID [4631]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 1
Problem number : 23
Date solved : Monday, January 27, 2025 at 09:27:21 AM
CAS classification : [_linear]

\begin{align*} y^{\prime }&=4 \csc \left (x \right ) x \left (1-\tan \left (x \right )^{2}+y\right ) \end{align*}

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 110

dsolve(diff(y(x),x) = 4*csc(x)*x*(1-tan(x)^2+y(x)),y(x), singsol=all)
 

\[ y \left (x \right ) = -4 \,{\mathrm e}^{-4 i \left (-\operatorname {dilog}\left ({\mathrm e}^{i x}+1\right )+\operatorname {dilog}\left (1-{\mathrm e}^{i x}\right )\right )} \left (1-{\mathrm e}^{i x}\right )^{4 x} \left ({\mathrm e}^{i x}+1\right )^{-4 x} \left (\int \csc \left (x \right ) \left (-2+\sec \left (x \right )^{2}\right ) x \left (1-{\mathrm e}^{i x}\right )^{-4 x} \left ({\mathrm e}^{i x}+1\right )^{4 x} {\mathrm e}^{4 i \left (-\operatorname {dilog}\left ({\mathrm e}^{i x}+1\right )+\operatorname {dilog}\left (1-{\mathrm e}^{i x}\right )\right )}d x -\frac {c_{1}}{4}\right ) \]

✓ Solution by Mathematica

Time used: 9.903 (sec). Leaf size: 156

DSolve[D[y[x],x]==2*Csc[x]*2*x*(1-Tan[x]^2+y[x]),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (4 i \operatorname {PolyLog}\left (2,-e^{i x}\right )-4 i \operatorname {PolyLog}\left (2,e^{i x}\right )+4 x \left (\log \left (1-e^{i x}\right )-\log \left (1+e^{i x}\right )\right )\right ) \left (\int _1^x4 \exp \left (4 K[1] \left (\log \left (1+e^{i K[1]}\right )-\log \left (1-e^{i K[1]}\right )\right )-4 i \operatorname {PolyLog}\left (2,-e^{i K[1]}\right )+4 i \operatorname {PolyLog}\left (2,e^{i K[1]}\right )\right ) \cos (2 K[1]) \csc (K[1]) K[1] \sec ^2(K[1])dK[1]+c_1\right ) \]