74.12.29 problem 29

Internal problem ID [16265]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 29
Date solved : Monday, March 31, 2025 at 02:48:50 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+4 y&={\mathrm e}^{2 t} \arctan \left (t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 38

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)+4*y(t) = exp(2*t)*arctan(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{2 t} \left (-\arctan \left (t \right ) t^{2}+t \ln \left (t^{2}+1\right )-2 t c_1 +\arctan \left (t \right )-2 c_2 -t \right )}{2} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 42

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

\[ y(t)\to \frac {1}{2} e^{2 t} \left (\left (t^2-1\right ) \arctan (t)-t \log \left (t^2+1\right )+t+2 c_2 t+2 c_1\right ) \]

✓ Sympy. Time used: 0.367 (sec). Leaf size: 32

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

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