89.13.12 problem 12

Internal problem ID [24595]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 8. Linear Differential Equations with constant coefficients. Exercises at page 127
Problem number : 12
Date solved : Thursday, October 02, 2025 at 10:46:23 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }+7 y^{\prime \prime }+19 y^{\prime }+13 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=2 \\ y^{\prime \prime }\left (0\right )&=-12 \\ \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 13

ode:=diff(diff(diff(y(x),x),x),x)+7*diff(diff(y(x),x),x)+19*diff(y(x),x)+13*y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 2, (D@@2)(y)(0) = -12]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-3 x} \sin \left (2 x \right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 15

ode=D[y[x],{x,3}]+7*D[y[x],{x,2}]+19*D[y[x],{x,1}]+13*y[x] ==0; 
ic={y[0]==0,Derivative[1][y][0] ==2,Derivative[2][y][0] ==-12}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-3 x} \sin (2 x) \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(13*y(x) + 19*Derivative(y(x), x) + 7*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 2, Subs(Derivative(y(x), (x, 2)), x, 0): -12} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = e^{- 3 x} \sin {\left (2 x \right )} \]