83.49.5 problem Ex 5 page 122

Internal problem ID [19618]
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 5 page 122
Date solved : Tuesday, January 28, 2025 at 08:33:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-\cot \left (x \right ) y^{\prime }-\left (1-\cot \left (x \right )\right ) y&={\mathrm e}^{x} \sin \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 27

dsolve(diff(y(x),x$2)-cot(x)*diff(y(x),x)-(1-cot(x))*y(x)=exp(x)*sin(x),y(x), singsol=all)
 

\[ y \left (x \right ) = \left (2 \sin \left (x \right )+\cos \left (x \right )\right ) c_{2} {\mathrm e}^{-x}+\left (c_{1} -\frac {\cos \left (x \right )}{2}\right ) {\mathrm e}^{x} \]

✓ Solution by Mathematica

Time used: 0.814 (sec). Leaf size: 52

DSolve[D[y[x],{x,2}]-Cot[x]*D[y[x],x]-(1-Cot[x])*y[x]==Exp[x]*Sin[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{10} e^{-x} \left (10 c_1 e^{2 x}-\left (5 e^{2 x}+2 c_2\right ) \cos \left (\log \left (e^x\right )\right )-4 c_2 \sin \left (\log \left (e^x\right )\right )\right ) \]