76.15.24 problem 25

Internal problem ID [17586]
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 : 25
Date solved : Thursday, March 13, 2025 at 10:16:05 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }+6 y&={\mathrm e}^{t} \cos \left (2 t \right )+{\mathrm e}^{2 t} \left (3 t +4\right ) \sin \left (t \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 50

ode:=diff(diff(y(t),t),t)-5*diff(y(t),t)+6*y(t) = exp(t)*cos(2*t)+exp(2*t)*(3*t+4)*sin(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {3 \,{\mathrm e}^{t} \left (\frac {2 c_{2} {\mathrm e}^{2 t}}{3}+\left (\left (t +\frac {1}{3}\right ) \cos \left (t \right )+\left (-t -\frac {10}{3}\right ) \sin \left (t \right )+\frac {2 c_{1}}{3}\right ) {\mathrm e}^{t}-\frac {\cos \left (t \right )^{2}}{15}-\frac {\sin \left (t \right ) \cos \left (t \right )}{5}+\frac {1}{30}\right )}{2} \]

✓ Mathematica. Time used: 0.58 (sec). Leaf size: 68

ode=D[y[t],{t,2}]-5*D[y[t],t]+6*y[t]==Exp[t]*Cos[2*t]+Exp[2*t]*(3*t+4)*Sin[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {1}{20} e^t \left (-100 e^t \sin (t)-30 e^t t \sin (t)-3 \sin (2 t)+10 e^t (3 t+1) \cos (t)-\cos (2 t)+20 c_1 e^t+20 c_2 e^{2 t}\right ) \]

✓ Sympy. Time used: 0.593 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-(3*t + 4)*exp(2*t)*sin(t) + 6*y(t) - exp(t)*cos(2*t) - 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{2} e^{2 t} + \left (C_{1} - \frac {3 t \sin {\left (t \right )}}{2} + \frac {3 t \cos {\left (t \right )}}{2} - 5 \sin {\left (t \right )} + \frac {\cos {\left (t \right )}}{2}\right ) e^{t} - \frac {3 \sin {\left (2 t \right )}}{20} - \frac {\cos {\left (2 t \right )}}{20}\right ) e^{t} \]