89.21.7 problem 9

Internal problem ID [24779]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Exercises at page 154
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:47:56 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\left (5\right )}-6 y^{\prime \prime \prime \prime }+12 y^{\prime \prime \prime }-8 y^{\prime \prime }&=48 \,{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 50

ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-6*diff(diff(diff(diff(y(x),x),x),x),x)+12*diff(diff(diff(y(x),x),x),x)-8*diff(diff(y(x),x),x) = 48*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (16 x^{3}+\left (2 c_3 -48\right ) x^{2}+\left (2 c_2 -4 c_3 +72\right ) x +2 c_1 -2 c_2 +3 c_3 -48\right ) {\mathrm e}^{2 x}}{8}+c_4 x +c_5 \]

✓ Mathematica. Time used: 0.194 (sec). Leaf size: 58

ode=D[y[x],{x,5}]-6*D[y[x],{x,4}]+12*D[y[x],{x,3}]-8*D[y[x],{x,2}]== 48*Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} e^{2 x} \left (8 x^3+(-24+c_3) x^2+(36+c_2-2 c_3) x-24+c_1-c_2+\frac {3 c_3}{2}\right )+c_5 x+c_4 \end{align*}

✓ Sympy. Time used: 0.123 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-48*exp(2*x) - 8*Derivative(y(x), (x, 2)) + 12*Derivative(y(x), (x, 3)) - 6*Derivative(y(x), (x, 4)) + Derivative(y(x), (x, 5)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{5} e^{2 x} + x \left (C_{2} + \left (C_{3} + x \left (C_{4} + 2 x\right )\right ) e^{2 x}\right ) \]