7.7.8 problem 8

Internal problem ID [2371]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2, linear equations with constant coefficients. Page 138
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 05:34:59 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.151 (sec). Leaf size: 37

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

\[ y = -\frac {{\mathrm e}^{-6+3 t} \left (-2 \cosh \left (2 \sqrt {2}\, \left (-2+t \right )\right )+\sqrt {2}\, \sinh \left (2 \sqrt {2}\, \left (-2+t \right )\right )\right )}{2} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 72

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

\begin{align*} y(t)&\to \frac {1}{4} e^{-6-4 \sqrt {2}} \left (\left (2+\sqrt {2}\right ) e^{\left (3-2 \sqrt {2}\right ) t+8 \sqrt {2}}-\left (\left (\sqrt {2}-2\right ) e^{\left (3+2 \sqrt {2}\right ) t}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.150 (sec). Leaf size: 92

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

\[ y{\left (t \right )} = \left (\frac {\sqrt {2} e^{4 \sqrt {2}}}{4 e^{6}} + \frac {e^{4 \sqrt {2}}}{2 e^{6}}\right ) e^{t \left (3 - 2 \sqrt {2}\right )} + \left (- \frac {\sqrt {2}}{4 e^{6} e^{4 \sqrt {2}}} + \frac {1}{2 e^{6} e^{4 \sqrt {2}}}\right ) e^{t \left (2 \sqrt {2} + 3\right )} \]