73.15.42 problem 22.11 (a)

Internal problem ID [15402]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 22. Method of undetermined coefficients. Additional exercises page 412
Problem number : 22.11 (a)
Date solved : Monday, March 31, 2025 at 01:36:40 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+5 y&=x^{3} {\mathrm e}^{-x} \sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 61

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+5*y(x) = x^3*exp(-x)*sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (39546 x^{3}+94302 x^{2}+95160 x +38200\right ) \cos \left (x \right )+59319 \sin \left (x \right ) \left (x^{3}+\frac {18}{13} x^{2}+\frac {138}{169} x +\frac {360}{2197}\right )\right ) {\mathrm e}^{-x}}{771147}+{\mathrm e}^{2 x} \left (\cos \left (x \right ) c_1 +\sin \left (x \right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 70

ode=D[y[x],{x,2}]-4*D[y[x],x]+5*y[x]==x^3*Exp[-x]*Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{-x} \left (\left (39546 x^3+94302 x^2+95160 x+771147 c_2 e^{3 x}+38200\right ) \cos (x)+27 \left (2197 x^3+3042 x^2+1794 x+28561 c_1 e^{3 x}+360\right ) \sin (x)\right )}{771147} \]

✓ Sympy. Time used: 0.574 (sec). Leaf size: 114

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3*exp(-x)*sin(x) + 5*y(x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{3} e^{- x} \sin {\left (x \right )}}{13} + \frac {2 x^{3} e^{- x} \cos {\left (x \right )}}{39} + \frac {18 x^{2} e^{- x} \sin {\left (x \right )}}{169} + \frac {62 x^{2} e^{- x} \cos {\left (x \right )}}{507} + \frac {138 x e^{- x} \sin {\left (x \right )}}{2197} + \frac {2440 x e^{- x} \cos {\left (x \right )}}{19773} + \left (C_{1} \sin {\left (x \right )} + C_{2} \cos {\left (x \right )}\right ) e^{2 x} + \frac {360 e^{- x} \sin {\left (x \right )}}{28561} + \frac {38200 e^{- x} \cos {\left (x \right )}}{771147} \]