69.17.2 problem 552

Internal problem ID [18338]
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. Superposition principle. Exercises page 137
Problem number : 552
Date solved : Thursday, October 02, 2025 at 03:10:36 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x) = x+exp(-4*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-4 x -4 c_1 -1\right ) {\mathrm e}^{-4 x}}{16}+\frac {x^{2}}{8}-\frac {x}{16}+c_2 \]

✓ Mathematica. Time used: 2.91 (sec). Leaf size: 44

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

\begin{align*} y(x)&\to \int _1^xe^{-4 K[2]} \left (c_1+\int _1^{K[2]}\left (e^{4 K[1]} K[1]+1\right )dK[1]\right )dK[2]+c_2 \end{align*}

✓ Sympy. Time used: 0.151 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = C_{1} + \frac {x^{2}}{8} - \frac {x}{16} + \left (C_{2} - \frac {x}{4}\right ) e^{- 4 x} \]