72.16.34 problem 35

Internal problem ID [14851]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 4. Forcing and Resonance. Section 4.1 page 399
Problem number : 35
Date solved : Thursday, March 13, 2025 at 04:21:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&=t -\frac {1}{20} t^{2} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+4*y(t) = t-1/20*t^2; 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -\frac {\sin \left (2 t \right )}{8}-\frac {\cos \left (2 t \right )}{160}-\frac {t^{2}}{80}+\frac {t}{4}+\frac {1}{160} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 31

ode=D[y[t],{t,2}]+4*y[t]==t-t^2/20; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{160} \left (-2 t^2+40 t-20 \sin (2 t)-\cos (2 t)+1\right ) \]

✓ Sympy. Time used: 0.106 (sec). Leaf size: 27

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

\[ y{\left (t \right )} = - \frac {t^{2}}{80} + \frac {t}{4} - \frac {\sin {\left (2 t \right )}}{8} - \frac {\cos {\left (2 t \right )}}{160} + \frac {1}{160} \]