83.49.24 problem Ex 24 page 139

Internal problem ID [19637]
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 24 page 139
Date solved : Tuesday, January 28, 2025 at 08:33:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\left (1-\cot \left (x \right )\right ) y^{\prime }-y \cot \left (x \right )&=\sin \left (x \right )^{2} \end{align*}

✓ Solution by Maple

Time used: 0.086 (sec). Leaf size: 82

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

\[ y = -\left (\int {\mathrm e}^{-\frac {\pi }{2}+x} \sin \left (x \right )d x \right ) \cos \left (x \right ) {\mathrm e}^{\frac {\pi }{2}-x}-{\mathrm e}^{-x} \left (\int {\mathrm e}^{-\frac {\pi }{2}+x} \sin \left (x \right )d x \right ) c_1 -\left (\int \left (\int {\mathrm e}^{-\frac {\pi }{2}+x} \sin \left (x \right )d x \right ) \sin \left (x \right )d x \right ) {\mathrm e}^{\frac {\pi }{2}-x}+{\mathrm e}^{\frac {\pi }{2}-x} c_2 \]

✓ Solution by Mathematica

Time used: 0.243 (sec). Leaf size: 53

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

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