64.12.26 problem 26

Internal problem ID [13459]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.4. Variation of parameters. Exercises page 162
Problem number : 26
Date solved : Monday, March 31, 2025 at 07:58:31 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 31

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

\[ y = c_2 \,{\mathrm e}^{-x}+c_3 \,{\mathrm e}^{3 x}-\frac {{\mathrm e}^{x} \left (x^{3}+\frac {3}{2} x -12 c_1 \right )}{12} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 92

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

\[ y(x)\to e^{-x} \int _1^x\frac {1}{8} e^{2 K[1]} K[1]^2dK[1]+e^{3 x} \int _1^x\frac {1}{8} e^{-2 K[2]} K[2]^2dK[2]-\frac {1}{12} e^x x^3+c_1 e^{-x}+c_2 e^x+c_3 e^{3 x} \]

✓ Sympy. Time used: 0.282 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{2} e^{- x} + C_{3} e^{3 x} + \left (C_{1} - \frac {x^{3}}{12} - \frac {x}{8}\right ) e^{x} \]