76.17.7 problem 16

Internal problem ID [17614]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.7 (Variation of parameters). Problems at page 280
Problem number : 16
Date solved : Thursday, March 13, 2025 at 10:43:46 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=\frac {{\mathrm e}^{2 t}}{t^{2}} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 36

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+4*y(t) = 1/t^2*exp(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (-4 t +1\right ) {\mathrm e}^{-2 t} \operatorname {Ei}_{1}\left (-4 t \right )+\left (c_{1} t +c_{2} \right ) {\mathrm e}^{-2 t}-{\mathrm e}^{2 t} \]

✓ Mathematica. Time used: 0.257 (sec). Leaf size: 92

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

\[ y(t)\to \frac {1}{2} e^{-\left (\left (2+\sqrt {2}\right ) t\right )} \left (\left (2 \sqrt {2}-1\right ) e^{2 \sqrt {2} t} \operatorname {ExpIntegralEi}\left (-\left (\left (-4+\sqrt {2}\right ) t\right )\right )-\left (1+2 \sqrt {2}\right ) \operatorname {ExpIntegralEi}\left (\left (4+\sqrt {2}\right ) t\right )+2 \left (c_2 e^{2 \sqrt {2} t}+c_1\right )\right ) \]

✓ Sympy. Time used: 0.864 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = \left (C_{1} + t \left (C_{2} + 4 \operatorname {Ei}{\left (4 t \right )}\right ) - \operatorname {Ei}{\left (4 t \right )}\right ) e^{- 2 t} - e^{2 t} \]