4.7.16 problem 16

Internal problem ID [1264]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.1 Homogeneous Equations with Constant Coefficients, page 144
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 04:31:51 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} y \left (-2\right )&=1 \\ y^{\prime }\left (-2\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.065 (sec). Leaf size: 21

ode:=4*diff(diff(y(x),x),x)-y(x) = 0; 
ic:=[y(-2) = 1, D(y)(-2) = -1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{1+\frac {x}{2}}}{2}+\frac {3 \,{\mathrm e}^{-1-\frac {x}{2}}}{2} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 25

ode=4*D[y[x],{x,2}]-y[x]==0; 
ic={y[-2]==1,Derivative[1][y][-2]==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.057 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + 4*Derivative(y(x), (x, 2)),0) 
ics = {y(-2): 1, Subs(Derivative(y(x), x), x, -2): -1} 
dsolve(ode,func=y(x),ics=ics)
 

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