75.18.26 problem 615

Internal problem ID [17043]
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 : 615
Date solved : Monday, March 31, 2025 at 03:39:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=2 \,{\mathrm e}^{x} \left (\sin \left (x \right )+7 \cos \left (x \right )\right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.177 (sec). Leaf size: 13

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = 2*exp(x)*(sin(x)+7*cos(x)); 
ic:=y(-infinity) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \operatorname {signum}\left ({\mathrm e}^{-2 x} c_1 \right ) \infty \]

✗ Mathematica

ode=D[y[x],{x,2}]+4*D[y[x],x]+4*y[x]==2*Exp[x]*(Sin[x]+7*Cos[x]); 
ic={y[-Infinity]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✓ Sympy. Time used: 0.315 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-2*sin(x) - 14*cos(x))*exp(x) + 4*y(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(-oo): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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