68.18.31 problem 37

Internal problem ID [17875]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 37
Date solved : Thursday, October 02, 2025 at 02:29:07 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+14 y^{\prime \prime }+20 y^{\prime }+25 y&=t^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 40

ode:=diff(diff(diff(diff(y(t),t),t),t),t)+4*diff(diff(diff(y(t),t),t),t)+14*diff(diff(y(t),t),t)+20*diff(y(t),t)+25*y(t) = t^2; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {4}{625}+\left (\left (c_3 t +c_1 \right ) \cos \left (2 t \right )+\sin \left (2 t \right ) \left (c_4 t +c_2 \right )\right ) {\mathrm e}^{-t}+\frac {t^{2}}{25}-\frac {8 t}{125} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 54

ode=D[y[t],{t,4}]+4*D[ y[t],{t,3}]+14*D[y[t],{t,2}]+20*D[y[t],t]+25*y[t]==t^2; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{625} \left (25 t^2-40 t+4\right )+e^{-t} (c_4 t+c_3) \cos (2 t)+e^{-t} (c_2 t+c_1) \sin (2 t) \end{align*}

✓ Sympy. Time used: 0.183 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2 + 25*y(t) + 20*Derivative(y(t), t) + 14*Derivative(y(t), (t, 2)) + 4*Derivative(y(t), (t, 3)) + Derivative(y(t), (t, 4)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t^{2}}{25} - \frac {8 t}{125} + \left (\left (C_{1} + C_{2} t\right ) \sin {\left (2 t \right )} + \left (C_{3} + C_{4} t\right ) \cos {\left (2 t \right )}\right ) e^{- t} + \frac {4}{625} \]