81.10.25 problem 14-25

Internal problem ID [21620]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 14. Second order homogeneous differential equations with constant coefficients. Page 297.
Problem number : 14-25
Date solved : Thursday, October 02, 2025 at 07:59:05 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 33

ode:=diff(diff(diff(y(x),x),x),x)-6*diff(diff(y(x),x),x)+11*diff(y(x),x)-6*y(x) = 0; 
ic:=[y(Pi) = 0, D(y)(Pi) = 0, (D@@2)(y)(Pi) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-\pi +x}}{2}-{\mathrm e}^{-2 \pi +2 x}+\frac {{\mathrm e}^{-3 \pi +3 x}}{2} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 27

ode=D[y[x],{x,3}]-6*D[y[x],{x,2}]+11*D[y[x],x]-6*y[x]==0; 
ic={y[Pi]==0,Derivative[1][y][Pi] ==0,Derivative[2][y][Pi] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} e^{x-3 \pi } \left (e^{\pi }-e^x\right )^2 \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-6*y(x) + 11*Derivative(y(x), x) - 6*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(pi): 0, Subs(Derivative(y(x), x), x, pi): 0, Subs(Derivative(y(x), (x, 2)), x, pi): 1} 
dsolve(ode,func=y(x),ics=ics)
 

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