86.6.5 problem 5

Internal problem ID [23138]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 5. Linear equations of the second order with constant coefficients. Exercise 5b at page 77
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:23:24 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+3 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y \left (\frac {\pi \sqrt {3}}{3}\right )&=5 \,{\mathrm e}^{-\frac {\pi \sqrt {3}}{2}} \\ \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 26

ode:=diff(diff(y(x),x),x)+3*diff(y(x),x)+3*y(x) = 0; 
ic:=[y(0) = 1, y(1/3*3^(1/2)*Pi) = 5*exp(-1/2*3^(1/2)*Pi)]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {3 x}{2}} \left (5 \sin \left (\frac {\sqrt {3}\, x}{2}\right )+\cos \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 38

ode=D[y[x],{x,2}]+3*D[y[x],x]+3*y[x]==0; 
ic={y[0]==1,y[Pi/Sqrt[3]]== 5*Exp[-Sqrt[3]/2*Pi] }; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-3 x/2} \left (5 \sin \left (\frac {\sqrt {3} x}{2}\right )+\cos \left (\frac {\sqrt {3} x}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.124 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*y(x) + 3*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, y(sqrt(3)*Pi/3): 5*exp(-sqrt(3)*Pi/2)} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\left (- \frac {\cos {\left (\frac {\Pi }{2} \right )}}{\sin {\left (\frac {\Pi }{2} \right )}} + \frac {5}{\sin {\left (\frac {\Pi }{2} \right )}}\right ) \sin {\left (\frac {\sqrt {3} x}{2} \right )} + \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {3 x}{2}} \]