75.18.6 problem 595

Internal problem ID [17023]
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 : 595
Date solved : Monday, March 31, 2025 at 03:38:24 PM
CAS classification : [[_2nd_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 13

ode:=diff(diff(y(x),x),x)+diff(y(x),x) = exp(-x); 
ic:=y(0) = 1, D(y)(0) = -1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -{\mathrm e}^{-x} x +1 \]

✓ Mathematica. Time used: 1.655 (sec). Leaf size: 45

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

\[ y(x)\to \int _1^xe^{-K[1]} (K[1]-1)dK[1]-\int _1^0e^{-K[1]} (K[1]-1)dK[1]+1 \]

✓ Sympy. Time used: 0.170 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = - x e^{- x} + 1 \]