76.17.12 problem 21

Internal problem ID [17627]
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 4.7 (Variation of parameters). Problems at page 280
Problem number : 21
Date solved : Monday, March 31, 2025 at 04:22:59 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 47

ode:=diff(diff(y(t),t),t)+4*y(t) = g(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 +\frac {\int \cos \left (2 t \right ) g \left (t \right )d t \sin \left (2 t \right )}{2}-\frac {\int \sin \left (2 t \right ) g \left (t \right )d t \cos \left (2 t \right )}{2} \]

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 67

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

\[ y(t)\to \cos (2 t) \int _1^t-\cos (K[1]) g(K[1]) \sin (K[1])dK[1]+\sin (2 t) \int _1^t\frac {1}{2} \cos (2 K[2]) g(K[2])dK[2]+c_1 \cos (2 t)+c_2 \sin (2 t) \]

✓ Sympy. Time used: 1.108 (sec). Leaf size: 39

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

\[ y{\left (t \right )} = \left (C_{1} - \frac {\int g{\left (t \right )} \sin {\left (2 t \right )}\, dt}{2}\right ) \cos {\left (2 t \right )} + \left (C_{2} + \frac {\int g{\left (t \right )} \cos {\left (2 t \right )}\, dt}{2}\right ) \sin {\left (2 t \right )} \]