75.16.72 problem 545

Internal problem ID [16966]
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. Trial and error method. Exercises page 132
Problem number : 545
Date solved : Thursday, March 13, 2025 at 09:03:06 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 30

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+y(x) = x^2*exp(-x)*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-x} \left (c_{2} +c_{1} x -\cos \left (x \right ) x^{2}+4 x \sin \left (x \right )+6 \cos \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 32

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

\[ y(x)\to e^{-x} \left (-\left (x^2-6\right ) \cos (x)+4 x \sin (x)+c_2 x+c_1\right ) \]

✓ Sympy. Time used: 0.359 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} - x \cos {\left (x \right )} + 4 \sin {\left (x \right )}\right ) + 6 \cos {\left (x \right )}\right ) e^{- x} \]