76.12.21 problem 33

Internal problem ID [17506]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 33
Date solved : Monday, March 31, 2025 at 04:16:04 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x y^{\prime \prime }-y^{\prime }+4 x^{3} 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:=x*diff(diff(y(x),x),x)-diff(y(x),x)+4*x^3*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: 0.015 (sec). Leaf size: 20

ode=x*D[y[x],{x,2}]-D[y[x],x]+4*x^3*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: 0.173 (sec). Leaf size: 20

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

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