14.11.7 problem 7

Internal problem ID [2600]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.5. Method of judicious guessing. Excercises page 164
Problem number : 7
Date solved : Sunday, March 30, 2025 at 12:11:13 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=t \sin \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 31

ode:=diff(diff(y(t),t),t)+4*y(t) = t*sin(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (-t^{2}+8 c_1 \right ) \cos \left (2 t \right )}{8}+\frac {\sin \left (2 t \right ) \left (t +16 c_2 \right )}{16} \]

✓ Mathematica. Time used: 0.098 (sec). Leaf size: 38

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

\[ y(t)\to \frac {1}{64} \left (\left (-8 t^2+1+64 c_1\right ) \cos (2 t)+4 (t+16 c_2) \sin (2 t)\right ) \]

✓ Sympy. Time used: 0.143 (sec). Leaf size: 24

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

\[ y{\left (t \right )} = \left (C_{1} - \frac {t^{2}}{8}\right ) \cos {\left (2 t \right )} + \left (C_{2} + \frac {t}{16}\right ) \sin {\left (2 t \right )} \]