50.3.5 problem 5

Internal problem ID [10149]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 07:05:14 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 (0\right )&=1 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 14

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

\[ y = \frac {3 \sin \left (x \right )}{2}-\frac {\cos \left (x \right ) x}{2} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 55

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

\begin{align*} y(x)&\to -\cos (x) \int _1^0-\sin ^2(K[1])dK[1]+\cos (x) \int _1^x-\sin ^2(K[1])dK[1]+\frac {11 \sin (x)}{8}-\frac {1}{8} \sin (3 x) \end{align*}

✓ Sympy. Time used: 0.060 (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 = {Subs(Derivative(y(x), x), x, 0): 1, y(0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {x \cos {\left (x \right )}}{2} + \frac {3 \sin {\left (x \right )}}{2} \]