70.15.27 problem 28

Internal problem ID [18955]
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 : 28
Date solved : Thursday, October 02, 2025 at 03:35:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=t^{2} \sin \left (2 t \right )+\left (6 t +7\right ) \cos \left (2 t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 41

ode:=diff(diff(y(t),t),t)+4*y(t) = t^2*sin(2*t)+(6*t+7)*cos(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (-8 t^{3}+96 c_1 +39 t +84\right ) \cos \left (2 t \right )}{96}+\frac {13 \sin \left (2 t \right ) \left (t^{2}+\frac {28}{13} t +\frac {16}{13} c_2 -\frac {1}{4}\right )}{16} \]

✓ Mathematica. Time used: 0.348 (sec). Leaf size: 107

ode=D[y[t],{t,2}]+4*y[t]==t^2*Sin[2*t]+(6*t+7)*Cos[2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \cos (2 t) \int _1^t\frac {1}{4} \left (-2 K[1]^2 \sin ^2(2 K[1])-(6 K[1]+7) \sin (4 K[1])\right )dK[1]+\sin (2 t) \int _1^t\frac {1}{4} \left (\sin (4 K[2]) K[2]^2+6 K[2]+\cos (4 K[2]) (6 K[2]+7)+7\right )dK[2]+c_1 \cos (2 t)+c_2 \sin (2 t) \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2*sin(2*t) - (6*t + 7)*cos(2*t) + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} + \frac {13 t^{2}}{16} + \frac {7 t}{4}\right ) \sin {\left (2 t \right )} + \left (C_{2} - \frac {t^{3}}{12} + \frac {13 t}{32}\right ) \cos {\left (2 t \right )} \]