14.11.12 problem 12

Internal problem ID [2605]
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 : 12
Date solved : Sunday, March 30, 2025 at 12:11:22 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+4 y&=t^{2}+\left (2 t +3\right ) \left (1+\cos \left (t \right )\right ) \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 58

ode:=diff(diff(y(t),t),t)+diff(y(t),t)+4*y(t) = t^2+(2*t+3)*(1+cos(t)); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {15}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {15}\, t}{2}\right ) c_1 +\frac {17}{32}+\frac {\left (6 t +5\right ) \cos \left (t \right )}{10}+\frac {\left (2 t +5\right ) \sin \left (t \right )}{10}+\frac {t^{2}}{4}+\frac {3 t}{8} \]

✓ Mathematica. Time used: 7.218 (sec). Leaf size: 87

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

\[ y(t)\to \frac {t^2}{4}+\frac {3 t}{8}+\frac {1}{5} t \sin (t)+\frac {\sin (t)}{2}+\frac {1}{10} (6 t+5) \cos (t)+c_2 e^{-t/2} \cos \left (\frac {\sqrt {15} t}{2}\right )+c_1 e^{-t/2} \sin \left (\frac {\sqrt {15} t}{2}\right )+\frac {17}{32} \]

✓ Sympy. Time used: 0.329 (sec). Leaf size: 70

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2 - (2*t + 3)*(cos(t) + 1) + 4*y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {t^{2}}{4} + \frac {t \sin {\left (t \right )}}{5} + \frac {3 t \cos {\left (t \right )}}{5} + \frac {3 t}{8} + \left (C_{1} \sin {\left (\frac {\sqrt {15} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {15} t}{2} \right )}\right ) e^{- \frac {t}{2}} + \frac {\sin {\left (t \right )}}{2} + \frac {\cos {\left (t \right )}}{2} + \frac {17}{32} \]