68.12.26 problem 26

Internal problem ID [17620]
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 : 26
Date solved : Thursday, October 02, 2025 at 02:26:21 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&={\mathrm e}^{-2 t} \sqrt {-t^{2}+1} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 38

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

\[ y = \frac {{\mathrm e}^{-2 t} \left (\left (t^{2}+2\right ) \sqrt {-t^{2}+1}+6 t c_1 +3 t \arcsin \left (t \right )+6 c_2 \right )}{6} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 57

ode=D[y[t],{t,2}]+4*D[y[t],t]+4*y[t]==Exp[-2*t]*Sqrt[1-t^2]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{6} e^{-2 t} \left (3 t \arcsin (t)+\sqrt {1-t^2} t^2+2 \sqrt {1-t^2}+6 c_2 t+6 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.251 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-sqrt(1 - t**2)*exp(-2*t) + 4*y(t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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