83.12.3 problem 3

Internal problem ID [22004]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter VII. Operational method. Ex. XV at page 121
Problem number : 3
Date solved : Thursday, October 02, 2025 at 08:21:36 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime }&=x^{3}+{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 45

ode:=diff(diff(diff(y(x),x),x),x)-6*diff(diff(y(x),x),x)+9*diff(y(x),x) = x^3+exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (3 c_1 x -c_1 +3 c_2 \right ) {\mathrm e}^{3 x}}{9}+\frac {x^{4}}{36}+\frac {2 x^{3}}{27}+\frac {x^{2}}{9}+\frac {8 x}{81}+c_3 +\frac {{\mathrm e}^{x}}{4} \]

✓ Mathematica. Time used: 0.198 (sec). Leaf size: 63

ode=D[y[x],{x,3}]-6*D[y[x],{x,2}]+9*D[y[x],x]==x^3+Exp[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^4}{36}+\frac {2 x^3}{27}+\frac {x^2}{9}+\frac {8 x}{81}+\frac {e^x}{4}+\frac {1}{9} e^{3 x} (c_2 (3 x-1)+3 c_1)+c_3 \end{align*}

✓ Sympy. Time used: 0.148 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**3 - exp(x) + 9*Derivative(y(x), x) - 6*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + \frac {x^{4}}{36} + \frac {2 x^{3}}{27} + \frac {x^{2}}{9} + \frac {8 x}{81} + \left (C_{2} + C_{3} x\right ) e^{3 x} + \frac {e^{x}}{4} \]