31.1.9 problem 10.2.11 (iii)

Internal problem ID [7684]
Book : Basic Training in Mathematics. By R. Shankar. Plenum Press. NY. 1995
Section : Chapter 10, Differential equations. Section 10.2, ODEs with constant Coefficients. page 307
Problem number : 10.2.11 (iii)
Date solved : Tuesday, September 30, 2025 at 04:55:54 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+16 y&=16 \cos \left (4 x \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 16

ode:=diff(diff(y(x),x),x)+16*y(x) = 16*cos(4*x); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \cos \left (4 x \right )+2 \sin \left (4 x \right ) x \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 86

ode=D[y[x],{x,2}]+16*y[x]==16*Cos[4*x]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sin (4 x) \int _1^04 \cos ^2(4 K[2])dK[2]+\sin (4 x) \int _1^x4 \cos ^2(4 K[2])dK[2]+\cos (4 x) \left (\int _1^x-2 \sin (8 K[1])dK[1]-\int _1^0-2 \sin (8 K[1])dK[1]+1\right ) \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(16*y(x) - 16*cos(4*x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 2 x \sin {\left (4 x \right )} + \cos {\left (4 x \right )} \]