72.16.40 problem 42

Internal problem ID [14857]
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 : 42
Date solved : Thursday, March 13, 2025 at 05:13:34 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=6+t^{2}+{\mathrm e}^{t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 27

ode:=diff(diff(y(t),t),t)+4*y(t) = 6+t^2+exp(t); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -\frac {\sin \left (2 t \right )}{10}-\frac {63 \cos \left (2 t \right )}{40}+\frac {11}{8}+\frac {t^{2}}{4}+\frac {{\mathrm e}^{t}}{5} \]

✓ Mathematica. Time used: 0.099 (sec). Leaf size: 127

ode=D[y[t],{t,2}]+4*y[t]==6+t^2+Exp[t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -\sin (2 t) \int _1^0\frac {1}{2} \cos (2 K[2]) \left (K[2]^2+e^{K[2]}+6\right )dK[2]+\sin (2 t) \int _1^t\frac {1}{2} \cos (2 K[2]) \left (K[2]^2+e^{K[2]}+6\right )dK[2]+\cos (2 t) \left (\int _1^t-\cos (K[1]) \left (K[1]^2+e^{K[1]}+6\right ) \sin (K[1])dK[1]-\int _1^0-\cos (K[1]) \left (K[1]^2+e^{K[1]}+6\right ) \sin (K[1])dK[1]\right ) \]

✓ Sympy. Time used: 0.129 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2 + 4*y(t) - exp(t) + Derivative(y(t), (t, 2)) - 6,0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t^{2}}{4} + \frac {e^{t}}{5} - \frac {\sin {\left (2 t \right )}}{10} - \frac {63 \cos {\left (2 t \right )}}{40} + \frac {11}{8} \]