76.15.26 problem 27

Internal problem ID [17596]
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 : 27
Date solved : Monday, March 31, 2025 at 04:19:56 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 48

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

\[ y = \frac {3}{4}+\frac {\left (2 t^{3}+3 c_1 t +3 c_2 \right ) {\mathrm e}^{2 t}}{3}+\frac {\left (2 t +1\right ) \cos \left (2 t \right )}{16}+\frac {t^{2}}{2}+t -\frac {\sin \left (2 t \right )}{16} \]

✓ Mathematica. Time used: 0.439 (sec). Leaf size: 67

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

\[ y(t)\to \frac {2}{3} e^{2 t} t^3+\frac {t^2}{2}+t+\frac {1}{16} (2 t+1) \cos (2 t)+c_2 e^{2 t} t+c_1 e^{2 t}-\frac {1}{8} \sin (t) \cos (t)+\frac {3}{4} \]

✓ Sympy. Time used: 0.361 (sec). Leaf size: 51

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

\[ y{\left (t \right )} = \frac {t^{2}}{2} + \frac {t \cos {\left (2 t \right )}}{8} + t + \left (C_{1} + t \left (C_{2} + \frac {2 t^{2}}{3}\right )\right ) e^{2 t} - \frac {\sin {\left (2 t \right )}}{16} + \frac {\cos {\left (2 t \right )}}{16} + \frac {3}{4} \]