23.3.237 problem 239

Internal problem ID [5951]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 239
Date solved : Tuesday, September 30, 2025 at 02:06:43 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 28

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

\[ y = {\mathrm e}^{-\frac {x}{4}} \left (c_1 \sin \left (\frac {\sqrt {3}\, x}{4}\right )+c_2 \cos \left (\frac {\sqrt {3}\, x}{4}\right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 42

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

\begin{align*} y(x)&\to e^{-x/4} \left (c_2 \cos \left (\frac {\sqrt {3} x}{4}\right )+c_1 \sin \left (\frac {\sqrt {3} x}{4}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {3} x}{4} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} x}{4} \right )}\right ) e^{- \frac {x}{4}} \]