58.11.48 problem 48

Internal problem ID [14779]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 48
Date solved : Thursday, October 02, 2025 at 09:53:27 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }+4 y^{\prime \prime }+3 y^{\prime }+y&=x^{2} {\mathrm e}^{-x}+3 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 72

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+3*diff(diff(diff(y(x),x),x),x)+4*diff(diff(y(x),x),x)+3*diff(y(x),x)+y(x) = x^2*exp(-x)+3*exp(-1/2*x)*cos(1/2*3^(1/2)*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-x} \left (-24-18 \,{\mathrm e}^{\frac {x}{2}} \left (x -\frac {2 c_3}{3}+\frac {1}{3}\right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )-6 \,{\mathrm e}^{\frac {x}{2}} \left (\left (x -5\right ) \sqrt {3}-2 c_4 \right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )+x^{4}+4 x^{3}+12 \left (-2+c_2 \right ) x +12 c_1 \right )}{12} \]

✓ Mathematica. Time used: 1.077 (sec). Leaf size: 104

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

\begin{align*} y(x)&\to \frac {1}{12} e^{-x} \left (x^4+4 x^3-24 x+12 c_4 x-6 e^{x/2} (3 x+1-2 c_2) \cos \left (\frac {\sqrt {3} x}{2}\right )-6 e^{x/2} \left (\sqrt {3} x-5 \sqrt {3}-2 c_1\right ) \sin \left (\frac {\sqrt {3} x}{2}\right )-24+12 c_3\right ) \end{align*}

✓ Sympy. Time used: 0.631 (sec). Leaf size: 63

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {x^{3}}{12} + \frac {x^{2}}{3}\right )\right ) e^{- x} + \left (\left (C_{3} - \frac {3 x}{2}\right ) \cos {\left (\frac {\sqrt {3} x}{2} \right )} + \left (C_{4} - \frac {\sqrt {3} x}{2}\right ) \sin {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} \]