50.3.9 problem 9

Internal problem ID [10153]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 07:05:17 PM
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 (2\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.190 (sec). Leaf size: 36

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

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 88

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

\begin{align*} y(x)&\to \frac {1}{8} \left (8 \cos (x) \int _1^x-\sin ^2(K[1])dK[1]-8 \cos (2) (\cos (x)+\tan (1) \sin (x)) \int _1^2-\sin ^2(K[1])dK[1]-2 \sin (4-x)+\sin (6-x)+2 \sin (x)-\sin (3 x)-\sin (x+2)+\sin (x+4)\right ) \end{align*}

✓ Sympy. Time used: 0.076 (sec). Leaf size: 82

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(2): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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