58.11.47 problem 47

Internal problem ID [14778]
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 : 47
Date solved : Thursday, October 02, 2025 at 09:53:27 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y&=x \,{\mathrm e}^{2 x}+x^{2} {\mathrm e}^{3 x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 39

ode:=diff(diff(diff(y(x),x),x),x)-6*diff(diff(y(x),x),x)+12*diff(y(x),x)-8*y(x) = x*exp(2*x)+x^2*exp(3*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x^{4}+24 \left (x^{2}-6 x +12\right ) {\mathrm e}^{x}+24 c_1 +24 c_2 x +24 c_3 \,x^{2}\right ) {\mathrm e}^{2 x}}{24} \]

✓ Mathematica. Time used: 0.082 (sec). Leaf size: 47

ode=D[y[x],{x,3}]-6*D[y[x],{x,2}]+12*D[y[x],x]-8*y[x]==x*Exp[2*x]+x^2*Exp[3*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.287 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*exp(3*x) - x*exp(2*x) - 8*y(x) + 12*Derivative(y(x), x) - 6*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + x \left (C_{3} + \frac {x^{2}}{24} + e^{x}\right ) - 6 e^{x}\right ) + 12 e^{x}\right ) e^{2 x} \]