66.16.30 problem 31

Internal problem ID [16213]
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 : 31
Date solved : Thursday, October 02, 2025 at 10:43:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+4 y&=-3 t^{2}+2 t +3 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 26

ode:=diff(diff(y(t),t),t)+4*y(t) = -3*t^2+2*t+3; 
ic:=[y(0) = 2, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {\sin \left (2 t \right )}{4}+\frac {7 \cos \left (2 t \right )}{8}-\frac {3 t^{2}}{4}+\frac {t}{2}+\frac {9}{8} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 31

ode=D[y[t],{t,2}]+4*y[t]==-3*t^2+2*t+3; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{8} \left (-6 t^2+4 t-2 \sin (2 t)-9 \cos (2 t)+9\right ) \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = - \frac {3 t^{2}}{4} + \frac {t}{2} - \frac {\sin {\left (2 t \right )}}{4} + \frac {7 \cos {\left (2 t \right )}}{8} + \frac {9}{8} \]