12.9.21 problem 21

Internal problem ID [1777]
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 : 21
Date solved : Saturday, March 29, 2025 at 11:39:07 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 4 x y^{\prime \prime }+2 y^{\prime }+y&=0 \end{align*}

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 17

ode:=4*x*diff(diff(y(x),x),x)+2*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 24

ode=4*x*D[y[x],{x,2}]+2*D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.179 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = \sqrt [4]{x} \left (C_{1} J_{\frac {1}{2}}\left (\sqrt {x}\right ) + C_{2} Y_{\frac {1}{2}}\left (\sqrt {x}\right )\right ) \]