50.1.89 problem 88

Internal problem ID [10087]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 88
Date solved : Tuesday, September 30, 2025 at 06:56:54 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.085 (sec). Leaf size: 29

ode:=diff(diff(y(x),x),x)+diff(y(x),x)+y(x) = 0; 
ic:=[D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{-\frac {x}{2}} \left (\sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\sin \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 44

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

\begin{align*} y(x)&\to c_1 e^{-x/2} \left (\sin \left (\frac {\sqrt {3} x}{2}\right )+\sqrt {3} \cos \left (\frac {\sqrt {3} x}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.099 (sec). Leaf size: 37

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

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