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]
\[ \boxed {y^{\prime }-4 \csc \left (x \right ) x \left (1-\tan \left (x \right )^{2}+y\right )=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 118
dsolve(diff(y(x),x) = 4*csc(x)*x*(1-tan(x)^2+y(x)),y(x), singsol=all)
\[ y \left (x \right ) = \left (1+{\mathrm e}^{i x}\right )^{-4 x} \left (1-{\mathrm e}^{i x}\right )^{4 x} \left (\int -4 \csc \left (x \right ) \left (\sec \left (x \right )^{2}-2\right ) x \left (1-{\mathrm e}^{i x}\right )^{-4 x} \left (1+{\mathrm e}^{i x}\right )^{4 x} {\mathrm e}^{-4 i \left (\operatorname {dilog}\left (1+{\mathrm e}^{i x}\right )-\operatorname {dilog}\left (1-{\mathrm e}^{i x}\right )\right )}d x +c_{1} \right ) {\mathrm e}^{4 i \left (\operatorname {dilog}\left (1+{\mathrm e}^{i x}\right )-\operatorname {dilog}\left (1-{\mathrm e}^{i x}\right )\right )} \]
✓ Solution by Mathematica
Time used: 10.973 (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 x \text {arctanh}\left (e^{i x}\right )-8 i \operatorname {PolyLog}\left (2,e^{i x}\right )+2 i \operatorname {PolyLog}\left (2,e^{2 i x}\right )\right ) \left (\int _1^x4 \exp \left (8 \text {arctanh}\left (e^{i K[1]}\right ) K[1]+8 i \operatorname {PolyLog}\left (2,e^{i K[1]}\right )-2 i \operatorname {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*}