58.11.45 problem 45

Internal problem ID [14776]
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 : 45
Date solved : Thursday, October 02, 2025 at 09:52:12 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+13 y&=x \,{\mathrm e}^{-3 x} \sin \left (2 x \right )+x^{2} {\mathrm e}^{-2 x} \sin \left (3 x \right ) \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 68

ode:=diff(diff(y(x),x),x)+6*diff(y(x),x)+13*y(x) = x*exp(-3*x)*sin(2*x)+x^2*exp(-2*x)*sin(3*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {3 \left (\left (\frac {13 x^{2}}{12}-\frac {26 c_1}{3}-\frac {39}{16}\right ) \cos \left (2 x \right )+\left (x^{2}-\frac {2}{13} x -\frac {180}{169}\right ) {\mathrm e}^{x} \cos \left (3 x \right )+\frac {2 \left (x^{2}-\frac {41}{13} x +\frac {563}{338}\right ) {\mathrm e}^{x} \sin \left (3 x \right )}{3}-\frac {13 \sin \left (2 x \right ) \left (x +16 c_2 \right )}{24}\right ) {\mathrm e}^{-3 x}}{26} \]

✓ Mathematica. Time used: 1.156 (sec). Leaf size: 82

ode=D[y[x],{x,2}]+6*D[y[x],x]+13*y[x]==x*Exp[-3*x]*Sin[2*x]+x^2*Exp[-2*x]*Sin[3*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{-3 x} \left (-32 e^x \left (338 x^2-1066 x+563\right ) \sin (3 x)-96 e^x \left (169 x^2-26 x-180\right ) \cos (3 x)-2197 \left (8 x^2-1-64 c_2\right ) \cos (2 x)+8788 (x+16 c_1) \sin (2 x)\right )}{140608} \end{align*}

✓ Sympy. Time used: 0.668 (sec). Leaf size: 92

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*exp(-2*x)*sin(3*x) - x*exp(-3*x)*sin(2*x) + 13*y(x) + 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (- \frac {x^{2} \sin {\left (3 x \right )}}{13} - \frac {3 x^{2} \cos {\left (3 x \right )}}{26} + \frac {41 x \sin {\left (3 x \right )}}{169} + \frac {3 x \cos {\left (3 x \right )}}{169} + \left (\left (C_{1} - \frac {x^{2}}{8}\right ) \cos {\left (2 x \right )} + \left (C_{2} + \frac {x}{16}\right ) \sin {\left (2 x \right )}\right ) e^{- x} - \frac {563 \sin {\left (3 x \right )}}{4394} + \frac {270 \cos {\left (3 x \right )}}{2197}\right ) e^{- 2 x} \]