71.13.12 problem 12

Internal problem ID [14461]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.2, page 248
Problem number : 12
Date solved : Monday, March 31, 2025 at 12:27:40 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y^{\prime }+6 y&=-2 \sin \left (3 x \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.211 (sec). Leaf size: 46

ode:=diff(diff(y(x),x),x)-diff(y(x),x)+6*y(x) = -2*sin(3*x); 
ic:=y(0) = 0, D(y)(0) = -1; 
dsolve([ode,ic],y(x),method='laplace');
 

\[ y = -\frac {13 \sin \left (\frac {\sqrt {23}\, x}{2}\right ) \sqrt {23}\, {\mathrm e}^{\frac {x}{2}}}{69}+\frac {\cos \left (\frac {\sqrt {23}\, x}{2}\right ) {\mathrm e}^{\frac {x}{2}}}{3}-\frac {\cos \left (3 x \right )}{3}+\frac {\sin \left (3 x \right )}{3} \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 67

ode=D[y[x],{x,2}]-D[y[x],x]+6*y[x]==-2*Sin[3*x]; 
ic={y[0]==0,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{69} \left (23 \sin (3 x)-13 \sqrt {23} e^{x/2} \sin \left (\frac {\sqrt {23} x}{2}\right )-23 \cos (3 x)+23 e^{x/2} \cos \left (\frac {\sqrt {23} x}{2}\right )\right ) \]

✓ Sympy. Time used: 0.254 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*y(x) + 2*sin(3*x) - Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): -1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (- \frac {13 \sqrt {23} \sin {\left (\frac {\sqrt {23} x}{2} \right )}}{69} + \frac {\cos {\left (\frac {\sqrt {23} x}{2} \right )}}{3}\right ) e^{\frac {x}{2}} + \frac {\sin {\left (3 x \right )}}{3} - \frac {\cos {\left (3 x \right )}}{3} \]