44.10.12 problem 3(a)

Internal problem ID [9267]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.2. THE METHOD OF UNDETERMINED COEFFICIENTS. Page 67
Problem number : 3(a)
Date solved : Tuesday, September 30, 2025 at 06:15:51 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=4 \cos \left (2 x \right )+6 \cos \left (x \right )+8 x^{2}-4 x \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

ode:=diff(diff(y(x),x),x)+4*y(x) = 4*cos(2*x)+6*cos(x)+8*x^2-4*x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -1+\frac {\left (1+4 c_1 \right ) \cos \left (2 x \right )}{4}+\left (c_2 +x \right ) \sin \left (2 x \right )+2 x^{2}-x +2 \cos \left (x \right ) \]

✓ Mathematica. Time used: 0.223 (sec). Leaf size: 103

ode=D[y[x],{x,2}]+4*y[x]==4*Cos[2*x]+6*Cos[x]+8*x^2-4*x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \cos (2 x) \int _1^x-\left (\left (4 K[1]^2-2 K[1]+3 \cos (K[1])+2 \cos (2 K[1])\right ) \sin (2 K[1])\right )dK[1]+\sin (2 x) \int _1^x\cos (2 K[2]) \left (4 K[2]^2-2 K[2]+3 \cos (K[2])+2 \cos (2 K[2])\right )dK[2]+c_1 \cos (2 x)+c_2 \sin (2 x) \end{align*}

✓ Sympy. Time used: 0.066 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x**2 + 4*x + 4*y(x) - 6*cos(x) - 4*cos(2*x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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