83.18.5 problem 4

Internal problem ID [22057]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Examination III. page 256
Problem number : 4
Date solved : Thursday, October 02, 2025 at 08:23:16 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ y^{\prime }\left (1\right )&=0 \\ y^{\prime \prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 20

ode:=diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)+3*diff(y(x),x)-y(x) = 1/x^3*exp(x); 
ic:=[y(1) = 0, D(y)(1) = 0, (D@@2)(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 24

ode=D[y[x],{x,3}]-3*D[y[x],{x,2}]+3*D[y[x],x]-y[x]==x^(-3)*Exp[x]; 
ic={y[1]==0,Derivative[1][y][1] ==0,Derivative[2][y][1] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} e^x \left (x^2-4 x+2 \log (x)+3\right ) \end{align*}

✓ Sympy. Time used: 0.237 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + 3*Derivative(y(x), x) - 3*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)) - exp(x)/x**3,0) 
ics = {y(1): 0, Subs(Derivative(y(x), x), x, 1): 0, Subs(Derivative(y(x), (x, 2)), x, 1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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