83.38.9 problem 9

Internal problem ID [19468]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (C) at page 133
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 01:45:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (3 \sin \left (x \right )-\cot \left (x \right )\right ) y^{\prime }+2 \sin \left (x \right )^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.284 (sec). Leaf size: 44

dsolve(diff(y(x),x$2)+(3*sin(x)-cot(x))*diff(y(x),x)+2*y(x)*sin(x)^2=0,y(x), singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{\frac {3 \cos \left (x \right ) \operatorname {csgn}\left (\csc \left (x \right )\right )}{2}} \left (c_{1} \sin \left (\frac {\cot \left (x \right )}{2 \sqrt {-\csc \left (x \right )^{2}}}\right )+c_{2} \cos \left (\frac {\cot \left (x \right )}{2 \sqrt {-\csc \left (x \right )^{2}}}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.129 (sec). Leaf size: 20

DSolve[D[y[x],{x,2}]+(3*Sin[x]-Cot[x])*D[y[x],x]+2*y[x]*Sin[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{\cos (x)} \left (c_2 e^{\cos (x)}+c_1\right ) \]