20.11.5 problem Problem 5

Internal problem ID [3787]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.9, Reduction of Order. page 572
Problem number : Problem 5
Date solved : Sunday, March 30, 2025 at 02:08:33 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\sin \left (x^{2}\right ) \end{align*}

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

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

\[ y = c_1 \sin \left (x^{2}\right )+c_2 \cos \left (x^{2}\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 20

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

\[ y(x)\to c_1 \cos \left (x^2\right )+c_2 \sin \left (x^2\right ) \]

✓ Sympy. Time used: 0.171 (sec). Leaf size: 20

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

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