70.15.20 problem 21

Internal problem ID [18948]
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 : 21
Date solved : Thursday, October 02, 2025 at 03:33:16 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 24

ode:=diff(diff(y(t),t),t)+4*y(t) = 3*sin(2*t); 
ic:=[y(0) = 2, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 108

ode=D[y[t],{t,2}]+4*y[t]==3*Sin[2*t]; 
ic={y[0]==2,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\sin (2 t) \int _1^0\frac {3}{4} \sin (4 K[2])dK[2]+\sin (2 t) \int _1^t\frac {3}{4} \sin (4 K[2])dK[2]+\cos (2 t) \left (-\int _1^0-\frac {3}{2} \sin ^2(2 K[1])dK[1]\right )+\cos (2 t) \int _1^t-\frac {3}{2} \sin ^2(2 K[1])dK[1]+2 \cos (2 t)-\sin (t) \cos (t) \end{align*}

✓ Sympy. Time used: 0.068 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = \left (2 - \frac {3 t}{4}\right ) \cos {\left (2 t \right )} - \frac {\sin {\left (2 t \right )}}{8} \]