73.9.21 problem 14.2 (k)

Internal problem ID [15282]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.2 (k)
Date solved : Tuesday, January 28, 2025 at 07:51:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \sin \left (x \right )^{2} y^{\prime \prime }-2 \cos \left (x \right ) \sin \left (x \right ) y^{\prime }+\left (1+\cos \left (x \right )^{2}\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\sin \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 12

dsolve([sin(x)^2*diff(y(x),x$2)-2*cos(x)*sin(x)*diff(y(x),x)+(1+cos(x)^2)*y(x)=0,sin(x)],singsol=all)
 

\[ y = \sin \left (x \right ) \left (c_{2} x +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 15

DSolve[Sin[x]^2*D[y[x],{x,2}]-2*Cos[x]*Sin[x]*D[y[x],x]+(1+Cos[x]^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to (c_2 x+c_1) \sin (x) \]