87.12.12 problem 13

Internal problem ID [23460]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 93
Problem number : 13
Date solved : Thursday, October 02, 2025 at 09:42:00 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=\sqrt {3} \\ \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x)+diff(y(x),x)+y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 3^(1/2)]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\frac {x}{2}} \left (\left (6+\sqrt {3}\right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )+3 \cos \left (\frac {\sqrt {3}\, x}{2}\right )\right )}{3} \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 49

ode=D[y[x],{x,2}]+D[y[x],{x,1}]+y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==Sqrt[3]}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} e^{-x/2} \left (\left (6+\sqrt {3}\right ) \sin \left (\frac {\sqrt {3} x}{2}\right )+3 \cos \left (\frac {\sqrt {3} x}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.100 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): sqrt(3)} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (x \left (-1 + \sqrt {3}\right ) + 1\right ) e^{x} \]