80.8.7 problem 7

Internal problem ID [18515]
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 : 7
Date solved : Monday, March 31, 2025 at 05:40:59 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }-y&=\sin \left (t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 34

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)-y(t) = sin(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{\left (-2+\sqrt {5}\right ) t} c_2 +{\mathrm e}^{-\left (2+\sqrt {5}\right ) t} c_1 -\frac {\cos \left (t \right )}{5}-\frac {\sin \left (t \right )}{10} \]

✓ Mathematica. Time used: 0.25 (sec). Leaf size: 47

ode=D[y[t],{t,2}]+4*D[y[t],t]-y[t]==Sin[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -\frac {\sin (t)}{10}-\frac {\cos (t)}{5}+e^{-\left (\left (2+\sqrt {5}\right ) t\right )} \left (c_2 e^{2 \sqrt {5} t}+c_1\right ) \]

✓ Sympy. Time used: 0.237 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t) - sin(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} e^{t \left (-2 + \sqrt {5}\right )} + C_{2} e^{- t \left (2 + \sqrt {5}\right )} - \frac {\sin {\left (t \right )}}{10} - \frac {\cos {\left (t \right )}}{5} \]