15.12.21 problem 21

Internal problem ID [3165]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 20, page 90
Problem number : 21
Date solved : Monday, January 27, 2025 at 07:24:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 77

dsolve(diff(y(x),x$2)+9*y(x)=csc(2*x),y(x), singsol=all)
 

\[ y = \frac {\sin \left (x \right ) \left (-1+4 \cos \left (x \right )^{2}\right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{6}+\frac {\left (4 \cos \left (x \right )^{3}-3 \cos \left (x \right )\right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{6}+4 \cos \left (x \right )^{3} c_{1} +4 \sin \left (x \right ) \cos \left (x \right )^{2} c_2 +\frac {\left (-9 c_{1} +4 \sin \left (x \right )\right ) \cos \left (x \right )}{3}-c_2 \sin \left (x \right ) \]

✓ Solution by Mathematica

Time used: 0.038 (sec). Leaf size: 65

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

\[ y(x)\to \frac {1}{6} \left (\cos (3 x) \text {arctanh}(\sin (x))+4 \sin (2 x)+\sin (3 x) \log \left (\sin \left (\frac {x}{2}\right )\right )+6 c_1 \cos (3 x)+6 c_2 \sin (3 x)-\sin (3 x) \log \left (\cos \left (\frac {x}{2}\right )\right )\right ) \]