2.11.19 problem 34

Internal problem ID [887]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 34
Date solved : Tuesday, September 30, 2025 at 04:19:03 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 17

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

\begin{align*} y(x)&\to \frac {1}{2} (x-2) \sin (x)+\cos (x) \end{align*}

✓ Sympy. Time used: 0.055 (sec). Leaf size: 14

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

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