50.3.7 problem 7

Internal problem ID [10151]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 7
Date solved : Tuesday, September 30, 2025 at 07:05:16 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 \\ \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 28

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

\[ y = \frac {\left (2 c_2 \cot \left (1\right )-x +1\right ) \cos \left (x \right )}{2}+\frac {\sin \left (x \right ) \left (2 c_2 +1\right )}{2} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 55

ode=D[y[x],{x,2}]+y[x]==Sin[x]; 
ic={Derivative[1][y][1] == 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]-\frac {1}{2} \sin (x) \cos ^2(x)+c_1 \cos (x)+\frac {1}{2} \sin (x) \left (\cos ^2(1)+2 c_1 \tan (1)\right ) \end{align*}

✓ Sympy. Time used: 0.066 (sec). Leaf size: 34

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} 
dsolve(ode,func=y(x),ics=ics)
 

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