83.49.14 problem Ex 14 page 131

Internal problem ID [19627]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter VIII. Linear equations of second order
Problem number : Ex 14 page 131
Date solved : Tuesday, January 28, 2025 at 02:05:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }+\cot \left (x \right ) y^{\prime }+\frac {\csc \left (x \right )^{2} y}{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 31

dsolve(diff(y(x),x$2)+cot(x)*diff(y(x),x)+1/2*csc(x)^2*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} \left (\csc \left (x \right )+\cot \left (x \right )\right )^{-\frac {i \sqrt {2}}{2}}+c_{2} \left (\csc \left (x \right )+\cot \left (x \right )\right )^{\frac {i \sqrt {2}}{2}} \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 33

DSolve[D[y[x],{x,2}]+Cot[x]*D[y[x],x]+1/2*Csc[x]^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \cos \left (\frac {\text {arctanh}(\cos (x))}{\sqrt {2}}\right )-c_2 \sin \left (\frac {\text {arctanh}(\cos (x))}{\sqrt {2}}\right ) \]