6.9.26 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 : Tuesday, September 30, 2025 at 05:19:17 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*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 14

ode:=4*x^2*sin(x)*diff(diff(y(x),x),x)-4*x*(x*cos(x)+sin(x))*diff(y(x),x)+(2*x*cos(x)+3*sin(x))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.152 (sec). Leaf size: 21

ode=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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt {\arccos (\cos (x))} (c_2 \cos (x)+c_1) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*sin(x)*Derivative(y(x), (x, 2)) - 4*x*(x*cos(x) + sin(x))*Derivative(y(x), x) + (2*x*cos(x) + 3*sin(x))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2*sin(x)*Derivative(y(x), (x, 2)) + x*y(x)*cos(x)/2 + 3*y(x)*sin(x)/4)/(x*(x*cos(x) + sin(x))) cannot be solved by the factorable group method