78.13.10 problem 1 (j)

Internal problem ID [18253]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 18. The Method of Undetermined Coefficients. Problems at page 132
Problem number : 1 (j)
Date solved : Thursday, March 13, 2025 at 11:51:02 AM
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: 0.004 (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);
 

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

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 28

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]
 

\[ y(x)\to -\frac {1}{2} e^x ((x-2 c_2) \cos (x)-2 c_1 \sin (x)) \]

✓ Sympy. Time used: 0.264 (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)
 

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