83.15.14 problem 5 (b)

Internal problem ID [22038]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter XV. The Laplace Transform. Ex. XXIII at page 251
Problem number : 5 (b)
Date solved : Thursday, October 02, 2025 at 08:21:53 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.074 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 47

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

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

✓ Sympy. Time used: 0.047 (sec). Leaf size: 49

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

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