64.11.19 problem 19

Internal problem ID [13398]
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 : 19
Date solved : Monday, March 31, 2025 at 07:53:11 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y&=x \,{\mathrm e}^{x}-4 \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{4 x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 41

ode:=diff(diff(diff(y(x),x),x),x)-6*diff(diff(y(x),x),x)+11*diff(y(x),x)-6*y(x) = x*exp(x)-4*exp(2*x)+6*exp(4*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\frac {7}{2}+4 c_3 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{3 x}+4 \left (4 x +c_2 \right ) {\mathrm e}^{x}+x^{2}+3 x +4 c_1 \right ) {\mathrm e}^{x}}{4} \]

✓ Mathematica. Time used: 0.262 (sec). Leaf size: 121

ode=D[y[x],{x,3}]-6*D[y[x],{x,2}]+11*D[y[x],x]-6*y[x]==x*Exp[x]-4*Exp[2*x]+6*Exp[4*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{4} e^x \left (4 e^x \int _1^x\left (-e^{-K[1]} K[1]-6 e^{2 K[1]}+4\right )dK[1]+4 e^{2 x} \int _1^x\frac {1}{2} e^{-2 K[2]} \left (K[2]-4 e^{K[2]}+6 e^{3 K[2]}\right )dK[2]+x^2-8 e^x+4 e^{3 x}+4 c_2 e^x+4 c_3 e^{2 x}+4 c_1\right ) \]

✓ Sympy. Time used: 0.346 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*exp(x) - 6*y(x) - 6*exp(4*x) + 4*exp(2*x) + 11*Derivative(y(x), x) - 6*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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