28.2.15 problem 15

Internal problem ID [4458]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 15
Date solved : Sunday, March 30, 2025 at 03:23:05 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 36

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

\[ y = \frac {\left (\left (-25 x +50 c_1 \right ) {\mathrm e}^{x}+20 x +28\right ) \cos \left (x \right )}{50}+\frac {\left (5 c_2 \,{\mathrm e}^{x}+x +\frac {2}{5}\right ) \sin \left (x \right )}{5} \]

✓ Mathematica. Time used: 0.309 (sec). Leaf size: 48

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

\[ y(x)\to \frac {1}{50} \left (\left (-5 \left (5 e^x-4\right ) x+50 c_2 e^x+28\right ) \cos (x)+2 \left (5 x+25 c_1 e^x+2\right ) \sin (x)\right ) \]

✓ Sympy. Time used: 0.369 (sec). Leaf size: 48

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

\[ y{\left (x \right )} = \frac {x \sin {\left (x \right )}}{5} + \frac {2 x \cos {\left (x \right )}}{5} + \left (C_{2} \sin {\left (x \right )} + \left (C_{1} - \frac {x}{2}\right ) \cos {\left (x \right )}\right ) e^{x} + \frac {2 \sin {\left (x \right )}}{25} + \frac {14 \cos {\left (x \right )}}{25} \]