78.2.37 problem 15.c

Internal problem ID [20989]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 2, Second order ODEs. Problems section 2.6
Problem number : 15.c
Date solved : Thursday, October 02, 2025 at 07:01:08 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=2 \cos \left (x \right )+1 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 12

ode:=diff(diff(y(x),x),x)+y(x) = 1+2*cos(x); 
ic:=[y(0) = 2, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \cos \left (x \right )+1+\sin \left (x \right ) x \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 13

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

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

✓ Sympy. Time used: 0.053 (sec). Leaf size: 12

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

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