80.8.6 problem 6

Internal problem ID [18514]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter VII. Linear equations of order higher than the first. section 56. Problems at page 163
Problem number : 6
Date solved : Monday, March 31, 2025 at 05:40:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} v^{\prime \prime }-6 v^{\prime }+13 v&={\mathrm e}^{-2 u} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 31

ode:=diff(diff(v(u),u),u)-6*diff(v(u),u)+13*v(u) = exp(-2*u); 
dsolve(ode,v(u), singsol=all);
 

\[ v = {\mathrm e}^{3 u} \sin \left (2 u \right ) c_2 +{\mathrm e}^{3 u} \cos \left (2 u \right ) c_1 +\frac {{\mathrm e}^{-2 u}}{29} \]

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 39

ode=D[v[u],{u,2}]-6*D[v[u],u]+13*v[u]==Exp[-2*u]; 
ic={}; 
DSolve[{ode,ic},v[u],u,IncludeSingularSolutions->True]
 

\[ v(u)\to \frac {e^{-2 u}}{29}+c_2 e^{3 u} \cos (2 u)+c_1 e^{3 u} \sin (2 u) \]

✓ Sympy. Time used: 0.249 (sec). Leaf size: 29

from sympy import * 
u = symbols("u") 
v = Function("v") 
ode = Eq(13*v(u) - 6*Derivative(v(u), u) + Derivative(v(u), (u, 2)) - exp(-2*u),0) 
ics = {} 
dsolve(ode,func=v(u),ics=ics)
 

\[ v{\left (u \right )} = \left (C_{1} \sin {\left (2 u \right )} + C_{2} \cos {\left (2 u \right )}\right ) e^{3 u} + \frac {e^{- 2 u}}{29} \]