83.18.3 problem 2

Internal problem ID [22055]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Examination III. page 256
Problem number : 2
Date solved : Thursday, October 02, 2025 at 08:23:15 PM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 34

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

\[ y = \frac {\left (x^{2}+2 c_1 -4 x +6\right ) {\mathrm e}^{x}}{2}-\frac {x^{3}}{2}-\frac {3 x^{2}}{2}+c_2 x +c_3 \]

✓ Mathematica. Time used: 0.171 (sec). Leaf size: 44

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

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

✓ Sympy. Time used: 0.076 (sec). Leaf size: 32

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

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