56.3.3 problem 3

Internal problem ID [8861]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 3
Date solved : Wednesday, March 05, 2025 at 06:56:50 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)+y(x) = sin(x); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 20

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

\[ y(x)\to -\frac {1}{2} x \cos (x)+\cos (x)+c_2 \sin (x) \]

✓ Sympy. Time used: 0.092 (sec). Leaf size: 15

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

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