12.9.26 problem 26

Internal problem ID [1782]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 26
Date solved : Monday, January 27, 2025 at 05:34:47 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \sin \left (x \right ) y^{\prime \prime }-4 x \left (x \cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime }+\left (2 x \cos \left (x \right )+3 \sin \left (x \right )\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 14

dsolve([4*x^2*sin(x)*diff(y(x),x$2)-4*x*(x*cos(x)+sin(x))*diff(y(x),x)+(2*x*cos(x)+3*sin(x))*y(x)=0,sqrt(x)],singsol=all)
 

\[ y = \sqrt {x}\, \left (c_1 +c_2 \cos \left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.246 (sec). Leaf size: 21

DSolve[4*x^2*Sin[x]*D[y[x],{x,2}]-4*x*(x*Cos[x]+Sin[x])*D[y[x],x]+(2*x*Cos[x]+3*Sin[x])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt {\arccos (\cos (x))} (c_2 \cos (x)+c_1) \]