49.13.3 problem 1(c)

Internal problem ID [7679]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 121
Problem number : 1(c)
Date solved : Sunday, March 30, 2025 at 12:18:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x^{2}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 18

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

\[ y(x)\to e^{x^2} (c_2 x+c_1) \]

✗ Sympy

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

False