80.1.4 problem 5

Internal problem ID [18534]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter 1. section 5. Problems at page 19
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 08:28:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime } \sin \left (x \right )+y^{\prime } \cos \left (x \right )+n y \sin \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 37

dsolve(diff(sin(x)*diff(y(x),x),x)+n*y(x)*sin(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} \operatorname {LegendreP}\left (\frac {\sqrt {4 n +1}}{2}-\frac {1}{2}, \cos \left (x \right )\right )+c_{2} \operatorname {LegendreQ}\left (\frac {\sqrt {4 n +1}}{2}-\frac {1}{2}, \cos \left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.112 (sec). Leaf size: 48

DSolve[D[Sin[x]*D[y[x],x],x]+n*y[x]*Sin[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {LegendreP}\left (\frac {1}{2} \left (\sqrt {4 n+1}-1\right ),\cos (x)\right )+c_2 \operatorname {LegendreQ}\left (\frac {1}{2} \left (\sqrt {4 n+1}-1\right ),\cos (x)\right ) \]