58.11.33 problem 33

Internal problem ID [14764]
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 : 33
Date solved : Thursday, October 02, 2025 at 09:50:55 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+13 y&=8 \sin \left (3 x \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 29

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+13*y(x) = 8*sin(3*x); 
ic:=[y(0) = 1, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (2 \,{\mathrm e}^{2 x}+3\right ) \cos \left (3 x \right )}{5}+\frac {2 \sin \left (3 x \right ) \cosh \left (x \right ) {\mathrm e}^{x}}{5} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 36

ode=D[y[x],{x,2}]-4*D[y[x],x]+13*y[x]==8*Sin[3*x]; 
ic={y[0]==1,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{5} \left (\left (e^{2 x}+1\right ) \sin (3 x)+\left (2 e^{2 x}+3\right ) \cos (3 x)\right ) \end{align*}

✓ Sympy. Time used: 0.164 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(13*y(x) - 8*sin(3*x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {\sin {\left (3 x \right )}}{5} + \frac {2 \cos {\left (3 x \right )}}{5}\right ) e^{2 x} + \frac {\sin {\left (3 x \right )}}{5} + \frac {3 \cos {\left (3 x \right )}}{5} \]