64.11.38 problem 38

Internal problem ID [13488]
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 : Tuesday, January 28, 2025 at 05:45:41 AM
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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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