58.11.38 problem 38

Internal problem ID [14769]
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 : Thursday, October 02, 2025 at 09:50:59 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y+y^{\prime \prime }&=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: 0.020 (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,op(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: 0.055 (sec). Leaf size: 19

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]
 

\begin{align*} y(x)&\to 3 \sin (x) \cos (x)-2 x \cos (2 x) \end{align*}

✓ Sympy. Time used: 0.063 (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)
 

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