75.18.23 problem 612

Internal problem ID [17040]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Initial value problem. Exercises page 140
Problem number : 612
Date solved : Monday, March 31, 2025 at 03:38:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&=4 \,{\mathrm e}^{-x} \end{align*}

With initial conditions

\begin{align*} y \left (\infty \right )&=0 \end{align*}

✓ Maple. Time used: 0.090 (sec). Leaf size: 12

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+y(x) = 4*exp(-x); 
ic:=y(infinity) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -\operatorname {signum}\left (c_1 \,{\mathrm e}^{x}\right ) \infty \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 10

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

\[ y(x)\to e^{-x} \]

✓ Sympy. Time used: 0.212 (sec). Leaf size: 17

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

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