88.22.6 problem 10

Internal problem ID [24166]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 160 (Laplace transform)
Problem number : 10
Date solved : Thursday, October 02, 2025 at 10:00:23 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-12 y^{\prime \prime \prime }+54 y^{\prime \prime }-108 y^{\prime }+81 y&=x^{2} {\mathrm e}^{3 x} \end{align*}

Using Laplace method

✓ Maple. Time used: 0.068 (sec). Leaf size: 67

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-12*diff(diff(diff(y(x),x),x),x)+54*diff(diff(y(x),x),x)-108*diff(y(x),x)+81*y(x) = x^2*exp(3*x); 
dsolve(ode,y(x),method='laplace');
 

\[ y = \frac {{\mathrm e}^{3 x} \left (x^{6}+60 x^{3} y^{\prime \prime \prime }\left (0\right )-180 y \left (0\right ) \left (9 x^{3}-9 x^{2}+6 x -2\right )+180 y^{\prime }\left (0\right ) x \left (9 x^{2}-6 x +2\right )-180 y^{\prime \prime }\left (0\right ) x^{2} \left (3 x -1\right )\right )}{360} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 41

ode=D[y[x],{x,4}]-12*D[y[x],{x,3}]+54*D[y[x],{x,2}]-108*D[y[x],{x,1}]+81*y[x]==x^2*Exp[3*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{360} e^{3 x} \left (x^6+360 c_4 x^3+360 c_3 x^2+360 c_2 x+360 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.303 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*exp(3*x) + 81*y(x) - 108*Derivative(y(x), x) + 54*Derivative(y(x), (x, 2)) - 12*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + x \left (C_{3} + x \left (C_{4} + \frac {x^{3}}{360}\right )\right )\right )\right ) e^{3 x} \]