52.1.30 problem 30

Internal problem ID [10401]
Book : Second order enumerated odes
Section : section 1
Problem number : 30
Date solved : Tuesday, September 30, 2025 at 07:23:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y&=\cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x)+diff(y(x),x)+y(x) = cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_2 +{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_1 +\sin \left (x \right ) \]

✓ Mathematica. Time used: 0.885 (sec). Leaf size: 140

ode=D[y[x],{x,2}]+D[y[x],x]+y[x]==Cos[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x/2} \left (\cos \left (\frac {\sqrt {3} x}{2}\right ) \int _1^x-\frac {2 e^{\frac {K[2]}{2}} \cos (K[2]) \sin \left (\frac {1}{2} \sqrt {3} K[2]\right )}{\sqrt {3}}dK[2]+\sin \left (\frac {\sqrt {3} x}{2}\right ) \int _1^x\frac {2 e^{\frac {K[1]}{2}} \cos (K[1]) \cos \left (\frac {1}{2} \sqrt {3} K[1]\right )}{\sqrt {3}}dK[1]+c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+c_1 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - cos(x) + Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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