69.18.7 problem 596

Internal problem ID [18382]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Initial value problem. Exercises page 140
Problem number : 596
Date solved : Thursday, October 02, 2025 at 03:11:11 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+9 y&=10 \sin \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)+6*diff(y(x),x)+9*y(x) = 10*sin(x); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {3 \,{\mathrm e}^{-3 x}}{5}+{\mathrm e}^{-3 x} x -\frac {3 \cos \left (x \right )}{5}+\frac {4 \sin \left (x \right )}{5} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 33

ode=D[y[x],{x,2}]+6*D[y[x],x]+9*y[x]==10*Sin[x]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{5} \left (5 e^{-3 x} x+3 e^{-3 x}+4 \sin (x)-3 \cos (x)\right ) \end{align*}

✓ Sympy. Time used: 0.140 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*y(x) - 10*sin(x) + 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (x + \frac {3}{5}\right ) e^{- 3 x} + \frac {4 \sin {\left (x \right )}}{5} - \frac {3 \cos {\left (x \right )}}{5} \]