82.54.10 problem Ex. 10

Internal problem ID [19026]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IX. Equations of the second order. problems at end of chapter at page 120
Problem number : Ex. 10
Date solved : Tuesday, January 28, 2025 at 12:46:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Solution by Maple

Time used: 0.155 (sec). Leaf size: 51

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

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

✓ Solution by Mathematica

Time used: 0.238 (sec). Leaf size: 16

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

\[ y(x)\to c_1 x-c_2 \cos (x) \]