89.14.15 problem 15

Internal problem ID [24619]
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 128
Problem number : 15
Date solved : Thursday, October 02, 2025 at 10:46:32 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }+6 y^{\prime \prime }+12 y^{\prime }+y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=1 \\ y^{\prime \prime }\left (0\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.089 (sec). Leaf size: 79

ode:=diff(diff(diff(y(x),x),x),x)+6*diff(diff(y(x),x),x)+12*diff(y(x),x)+y(x) = 0; 
ic:=[y(0) = 1, D(y)(0) = 1, (D@@2)(y)(0) = -2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {2 \left (\left (7^{{1}/{3}}+\frac {7^{{2}/{3}}}{2}-\frac {7}{3}\right ) \cos \left (\frac {\sqrt {3}\, 7^{{1}/{3}} x}{2}\right )+\sqrt {3}\, \left (7^{{1}/{3}}-\frac {7^{{2}/{3}}}{2}\right ) \sin \left (\frac {\sqrt {3}\, 7^{{1}/{3}} x}{2}\right )-{\mathrm e}^{\frac {3 x 7^{{1}/{3}}}{2}} \left (7^{{1}/{3}}+\frac {7^{{2}/{3}}}{2}+\frac {7}{6}\right )\right ) {\mathrm e}^{-\frac {\left (7^{{1}/{3}}+4\right ) x}{2}}}{7} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 1108

ode=D[y[x],{x,3}]+6*D[y[x],{x,2}]+12*D[y[x],{x,1}]+y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==1,Derivative[2][y][0] ==-2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✓ Sympy. Time used: 0.307 (sec). Leaf size: 124

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) + 12*Derivative(y(x), x) + 6*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1, Subs(Derivative(y(x), (x, 2)), x, 0): -2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {7^{\frac {2}{3}}}{21} + \frac {2 \sqrt [3]{7}}{21}\right ) e^{x \left (-2 + \sqrt [3]{7}\right )} + \left (- \frac {2 \sqrt {3} \sqrt [3]{7}}{21} + \frac {\sqrt {3} \cdot 7^{\frac {2}{3}}}{21}\right ) e^{- x \left (\frac {\sqrt [3]{7}}{2} + 2\right )} \sin {\left (\frac {\sqrt {3} \sqrt [3]{7} x}{2} \right )} + \left (- \frac {2 \sqrt [3]{7}}{21} - \frac {7^{\frac {2}{3}}}{21}\right ) e^{- x \left (\frac {\sqrt [3]{7}}{2} + 2\right )} \cos {\left (\frac {\sqrt {3} \sqrt [3]{7} x}{2} \right )} \]