64.11.38 problem 38

Internal problem ID [13409]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 38
Date solved : Wednesday, March 05, 2025 at 09:52:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=8 \sin \left (2 x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=6\\ y^{\prime }\left (0\right )&=8 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)+4*y(x) = 8*sin(2*x); 
ic:=y(0) = 6, D(y)(0) = 8; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \left (-2 x +6\right ) \cos \left (2 x \right )+5 \sin \left (2 x \right ) \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 91

ode=D[y[x],{x,2}]+4*y[x]==8*Sin[2*x]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} \sin (2 x) \left (2 \int _1^x2 \sin (4 K[2])dK[2]-2 \int _1^02 \sin (4 K[2])dK[2]+1\right )-\cos (2 x) \int _1^0-4 \sin ^2(2 K[1])dK[1]+\cos (2 x) \int _1^x-4 \sin ^2(2 K[1])dK[1] \]

✓ Sympy. Time used: 0.106 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - 8*sin(2*x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 6, Subs(Derivative(y(x), x), x, 0): 8} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (6 - 2 x\right ) \cos {\left (2 x \right )} + 5 \sin {\left (2 x \right )} \]