44.10.10 problem 1(j)

Internal problem ID [9265]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.2. THE METHOD OF UNDETERMINED COEFFICIENTS. Page 67
Problem number : 1(j)
Date solved : Tuesday, September 30, 2025 at 06:15:50 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 51

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

\begin{align*} y(x)&\to \frac {1}{2} e^x \left (2 \cos (x) \int _1^x-\sin ^2(K[1])dK[1]-\sin (x) \cos ^2(x)+2 c_2 \cos (x)+2 c_1 \sin (x)\right ) \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \left (C_{2} \sin {\left (x \right )} + \left (C_{1} - \frac {x}{2}\right ) \cos {\left (x \right )}\right ) e^{x} \]