77.1.135 problem 162 (page 236)

Internal problem ID [18025]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 162 (page 236)
Date solved : Tuesday, January 28, 2025 at 11:20:49 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=\ln \left (2 \sin \left (\frac {x}{2}\right )\right ) \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 128

dsolve(diff(y(x),x$2)+9*y(x)=ln( 2*sin(x/2) ),y(x), singsol=all)
 

\[ y = -\frac {1}{54}+\frac {i \pi \left (\operatorname {csgn}\left (\sin \left (\frac {x}{2}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-\frac {i x}{2}}\right )+1\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{i x}-1\right )\right )}{18}+\frac {i \operatorname {csgn}\left (i {\mathrm e}^{-\frac {i x}{2}}\right ) \pi }{18}+\frac {i \pi \left (\operatorname {csgn}\left (\sin \left (\frac {x}{2}\right )\right )-1\right ) \operatorname {csgn}\left (i \sin \left (\frac {x}{2}\right )\right )}{18}+\frac {\left (1-\cos \left (3 x \right )\right ) \ln \left ({\mathrm e}^{i x}-1\right )}{9}-\frac {\ln \left ({\mathrm e}^{\frac {i x}{2}}\right )}{9}-\frac {{\mathrm e}^{-i x}}{36}-\frac {{\mathrm e}^{i x}}{36}+\frac {\cos \left (3 x \right ) \left (i x +18 c_{1} \right )}{18}+\frac {\left (-x +18 c_{2} \right ) \sin \left (3 x \right )}{18}-\frac {\cos \left (2 x \right )}{9} \]

✓ Solution by Mathematica

Time used: 0.107 (sec). Leaf size: 76

DSolve[D[y[x],{x,2}]+9*y[x]==Log[2*Sin[x/2]],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{54} \left (-3 x \sin (3 x)-3 \cos (x)-6 \cos (2 x)+10 \cos (3 x)+6 \log \left (2 \sin \left (\frac {x}{2}\right )\right )+54 c_1 \cos (3 x)+54 c_2 \sin (3 x)-6 \cos (3 x) \log \left (2 \sin \left (\frac {x}{2}\right )\right )-1\right ) \]