56.3.11 problem 11

Internal problem ID [8869]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 11
Date solved : Sunday, March 30, 2025 at 01:45:07 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\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: 0.501 (sec). Leaf size: 145

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

\[ y = \frac {2 \sin \left (1\right ) \left (\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sin \left (\sqrt {3}\right )-\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \cos \left (\sqrt {3}\right )\right ) {\mathrm e}^{\frac {1}{2}-\frac {x}{2}}-\left (\left (-\sqrt {3}\, \cos \left (\frac {\sqrt {3}}{2}\right )+\sin \left (\frac {\sqrt {3}}{2}\right )\right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )-\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}}{2}\right )+\cos \left (\frac {\sqrt {3}}{2}\right )\right )\right ) \cos \left (2\right ) {\mathrm e}^{1-\frac {x}{2}}-\left (\sqrt {3}\, \cos \left (\frac {\sqrt {3}}{2}\right )+\sin \left (\frac {\sqrt {3}}{2}\right )\right ) \cos \left (x \right )}{\sqrt {3}\, \cos \left (\frac {\sqrt {3}}{2}\right )+\sin \left (\frac {\sqrt {3}}{2}\right )} \]

✓ Mathematica. Time used: 0.866 (sec). Leaf size: 12765

ode=D[y[x],{x,3}]+D[y[x],x]+y[x]==Sin[x]; 
ic={Derivative[1][y][1] == 0,y[2]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✓ Sympy. Time used: 0.608 (sec). Leaf size: 2966

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - sin(x) + Derivative(y(x), 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)
 

\[ \text {Solution too large to show} \]