44.14.20 problem 3(d)

Internal problem ID [9344]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Problems for Review and Discovery. Drill excercises. Page 105
Problem number : 3(d)
Date solved : Tuesday, September 30, 2025 at 06:16:43 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 44

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

\[ y = {\mathrm e}^{-\left (1+\sqrt {2}\right ) x} c_1 +{\mathrm e}^{\left (\sqrt {2}-1\right ) x} c_2 -\frac {4 \left (\left (x -\frac {31}{34}\right ) \cos \left (x \right )-\frac {\sin \left (x \right ) \left (x +\frac {44}{17}\right )}{4}\right ) {\mathrm e}^{x}}{17} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 59

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

\begin{align*} y(x)&\to c_1 e^{-\left (\left (1+\sqrt {2}\right ) x\right )}+c_2 e^{\left (\sqrt {2}-1\right ) x}+\frac {1}{289} e^x ((17 x+44) \sin (x)+(62-68 x) \cos (x)) \end{align*}

✓ Sympy. Time used: 0.219 (sec). Leaf size: 68

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

\[ y{\left (x \right )} = C_{1} e^{x \left (-1 + \sqrt {2}\right )} + C_{2} e^{- x \left (1 + \sqrt {2}\right )} + \frac {x e^{x} \sin {\left (x \right )}}{17} - \frac {4 x e^{x} \cos {\left (x \right )}}{17} + \frac {44 e^{x} \sin {\left (x \right )}}{289} + \frac {62 e^{x} \cos {\left (x \right )}}{289} \]