58.10.42 problem 42

Internal problem ID [14729]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.2. The homogeneous linear equation with constant coefficients. Exercises page 135
Problem number : 42
Date solved : Thursday, October 02, 2025 at 09:50:34 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-5 y^{\prime \prime }+9 y^{\prime }-5 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ y^{\prime \prime }\left (0\right )&=6 \\ \end{align*}

✓ Maple. Time used: 0.085 (sec). Leaf size: 21

ode:=diff(diff(diff(y(x),x),x),x)-5*diff(diff(y(x),x),x)+9*diff(y(x),x)-5*y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = 6]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 25

ode=D[y[x],{x,3}]-5*D[y[x],{x,2}]+9*D[y[x],x]-5*y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1,Derivative[2][y][0] ==6}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x \left (2 e^x \sin (x)-e^x \cos (x)+1\right ) \end{align*}

✓ Sympy. Time used: 0.136 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-5*y(x) + 9*Derivative(y(x), x) - 5*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): 6} 
dsolve(ode,func=y(x),ics=ics)
 

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