56.3.8 problem 8

Internal problem ID [8866]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 8
Date solved : Wednesday, March 05, 2025 at 06:57:08 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^{\prime }\left (1\right )&=0\\ y \left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 23

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

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

✓ Sympy. Time used: 0.117 (sec). Leaf size: 24

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

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