76.13.43 problem 43

Internal problem ID [17554]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.3 (Linear homogeneous equations with constant coefficients). Problems at page 239
Problem number : 43
Date solved : Monday, March 31, 2025 at 04:17:17 PM
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: 0.061 (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,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.011 (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]
 

\[ y(x)\to -\frac {1}{2} e^{-\frac {x}{2}-1} \left (e^{x+2}-3\right ) \]

✓ Sympy. Time used: 0.118 (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)
 

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