65.11.1 problem 1

Internal problem ID [15797]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.4, page 218
Problem number : 1
Date solved : Thursday, October 02, 2025 at 10:28:07 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+13 y^{\prime \prime }-12 y^{\prime }+4 y&=2 \,{\mathrm e}^{x}-4 \,{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 36

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-6*diff(diff(diff(y(x),x),x),x)+13*diff(diff(y(x),x),x)-12*diff(y(x),x)+4*y(x) = 2*exp(x)-4*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} \left (c_3 x +x^{2}+c_1 +4 x +6+\left (c_4 x -2 x^{2}+c_2 +8 x -12\right ) {\mathrm e}^{x}\right ) \]

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 103

ode=D[y[x],{x,4}]-6*D[y[x],{x,3}]+13*D[y[x],{x,2}]-12*D[y[x],x]+4*y[x]==2*Exp[x]-4*Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x \left (\int _1^x2 \left (-1+2 e^{K[1]}\right ) (K[1]-2)dK[1]+e^x \int _1^x2 e^{-K[2]} \left (-1+2 e^{K[2]}\right ) (K[2]+2)dK[2]-4 e^x x^2+2 x^2-4 e^x x-2 x+c_2 x+c_4 e^x x+c_3 e^x+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.217 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) + 4*exp(2*x) - 2*exp(x) - 12*Derivative(y(x), x) + 13*Derivative(y(x), (x, 2)) - 6*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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