50.3.10 problem 10

Internal problem ID [10154]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 10
Date solved : Tuesday, September 30, 2025 at 07:05:18 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 (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.040 (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,op(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: 0.016 (sec). Leaf size: 62

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]
 

\begin{align*} y(x)&\to \cos (x) \int _1^x-\sin ^2(K[1])dK[1]-(\cos (x)+\tan (1) \sin (x)) \int _1^0-\sin ^2(K[1])dK[1]+\frac {1}{4} \sin (x) (\cos (2)-\cos (2 x)) \end{align*}

✓ Sympy. Time used: 0.066 (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)
 

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