29.1.23 problem 22

Internal problem ID [4630]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 1
Problem number : 22
Date solved : Monday, January 27, 2025 at 09:27:09 AM
CAS classification : [_linear]

\begin{align*} y^{\prime }&=4 \csc \left (x \right ) x \left (\sin \left (x \right )^{3}+y\right ) \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 108

dsolve(diff(y(x),x) = 4*csc(x)*x*(sin(x)^3+y(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 6.941 (sec). Leaf size: 148

DSolve[D[y[x],x]==2*Csc[x]*2*x(Sin[x]^3+y[x]),y[x],x,IncludeSingularSolutions -> True]
 

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