71.1.131 problem 158 (page 236)

Internal problem ID [19307]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 158 (page 236)
Date solved : Thursday, October 02, 2025 at 04:18:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y&={\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 38

ode:=diff(diff(y(x),x),x)+diff(y(x),x)+y(x) = exp(-1/2*x)*sin(1/2*3^(1/2)*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.138 (sec). Leaf size: 60

ode=D[y[x],{x,2}]+D[y[x],x]+y[x]==Exp[-x/2]*Sin[x*Sqrt[3]/2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.218 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) + Derivative(y(x), x) + Derivative(y(x), (x, 2)) - exp(-x/2)*sin(sqrt(3)*x/2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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