52.3.8 problem 8

Internal problem ID [8294]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.4 SPECIAL FUNCTIONS. EXERCISES 6.4. Page 267
Problem number : 8
Date solved : Wednesday, March 05, 2025 at 05:34:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (36 x^{2}-\frac {1}{4}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 21

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(36*x^2-1/4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{1} \sin \left (6 x \right )+c_{2} \cos \left (6 x \right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.064 (sec). Leaf size: 39

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(36*x^2-1/4)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{-6 i x} \left (12 c_1-i c_2 e^{12 i x}\right )}{12 \sqrt {x}} \]

✓ Sympy. Time used: 0.248 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (36*x**2 - 1/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} J_{\frac {1}{2}}\left (6 x\right ) + C_{2} Y_{\frac {1}{2}}\left (6 x\right ) \]