50.1.43 problem 43

Internal problem ID [10041]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 43
Date solved : Tuesday, September 30, 2025 at 06:50:55 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+4 y&=\sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

ode:=diff(diff(y(x),x),x)+diff(y(x),x)+4*y(x) = sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {15}\, x}{2}\right ) c_2 +{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {15}\, x}{2}\right ) c_1 +\frac {3 \sin \left (x \right )}{10}-\frac {\cos \left (x \right )}{10} \]

✓ Mathematica. Time used: 1.662 (sec). Leaf size: 140

ode=D[y[x],{x,2}]+D[y[x],x]+4*y[x]==Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x/2} \left (\cos \left (\frac {\sqrt {15} x}{2}\right ) \int _1^x-\frac {2 e^{\frac {K[2]}{2}} \sin (K[2]) \sin \left (\frac {1}{2} \sqrt {15} K[2]\right )}{\sqrt {15}}dK[2]+\sin \left (\frac {\sqrt {15} x}{2}\right ) \int _1^x\frac {2 e^{\frac {K[1]}{2}} \cos \left (\frac {1}{2} \sqrt {15} K[1]\right ) \sin (K[1])}{\sqrt {15}}dK[1]+c_2 \cos \left (\frac {\sqrt {15} x}{2}\right )+c_1 \sin \left (\frac {\sqrt {15} x}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - sin(x) + Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {15} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {15} x}{2} \right )}\right ) e^{- \frac {x}{2}} + \frac {3 \sin {\left (x \right )}}{10} - \frac {\cos {\left (x \right )}}{10} \]