29.1.21 problem 20

Internal problem ID [4628]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 1
Problem number : 20
Date solved : Monday, January 27, 2025 at 09:26:47 AM
CAS classification : [_linear]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 82

dsolve(diff(y(x),x) = 4*csc(x)*x*sec(x)^2-2*y(x)*cot(2*x),y(x), singsol=all)
 

\[ y \left (x \right ) = 16 \,\operatorname {csgn}\left (\csc \left (2 x \right )\right ) \left (\left (i \operatorname {dilog}\left (1+i {\mathrm e}^{i x}\right )-i \operatorname {dilog}\left (1-i {\mathrm e}^{i x}\right )-x \ln \left (1+i {\mathrm e}^{i x}\right )+x \ln \left (1-i {\mathrm e}^{i x}\right )\right ) \sqrt {-\frac {{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{4 i x}-1\right )^{2}}}+\frac {\csc \left (2 x \right ) c_{1}}{16}\right ) \]

✓ Solution by Mathematica

Time used: 0.077 (sec). Leaf size: 60

DSolve[D[y[x],x]==2*Csc[x]*2*x*Sec[x]^2-2*y[x]*Cot[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \csc (x) \sec (x) \left (-8 i x \arctan \left (e^{i x}\right )+4 i \operatorname {PolyLog}\left (2,-i e^{i x}\right )-4 i \operatorname {PolyLog}\left (2,i e^{i x}\right )+c_1\right ) \]