76.15.28 problem 29

Internal problem ID [17590]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 29
Date solved : Thursday, March 13, 2025 at 10:34:00 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 73

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

\[ y = -\frac {5 \left (\left (\left (t^{2}-\frac {73}{65} t +\frac {821}{676}\right ) \cos \left (t \right )^{2}-\frac {\sin \left (t \right ) \left (t^{2}+\frac {40}{13} t -\frac {1233}{676}\right ) \cos \left (t \right )}{5}-\frac {t^{2}}{2}+\frac {73 t}{130}-\frac {82619}{20280}\right ) {\mathrm e}^{3 t}-\frac {26 c_{2} {\mathrm e}^{t}}{5}+\frac {39 \,{\mathrm e}^{t} \cos \left (t \right )}{5}-\frac {39 \,{\mathrm e}^{t} \sin \left (t \right )}{5}+\frac {26 c_{1}}{5}\right ) {\mathrm e}^{-2 t}}{26} \]

✓ Mathematica. Time used: 0.617 (sec). Leaf size: 109

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

\[ y(t)\to \frac {1}{52} e^t t^2 \sin (2 t)-\frac {e^t \left (3380 t^2-3796 t+4105\right ) \cos (2 t)}{35152}+\frac {2 e^t}{3}+\frac {10}{169} e^t t \sin (2 t)+\frac {3}{2} e^{-t} \sin (t)-\frac {1233 e^t \sin (2 t)}{35152}-\frac {3}{2} e^{-t} \cos (t)+c_1 e^{-2 t}+c_2 e^{-t} \]

✓ Sympy. Time used: 0.741 (sec). Leaf size: 85

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

\[ y{\left (t \right )} = C_{2} e^{- 2 t} + \left (C_{1} + \frac {3 \sin {\left (t \right )}}{2} - \frac {3 \cos {\left (t \right )}}{2}\right ) e^{- t} + \frac {\left (2028 t^{2} \sin {\left (2 t \right )} - 10140 t^{2} \cos {\left (2 t \right )} + 6240 t \sin {\left (2 t \right )} + 11388 t \cos {\left (2 t \right )} - 3699 \sin {\left (2 t \right )} - 12315 \cos {\left (2 t \right )} + 70304\right ) e^{t}}{105456} \]