23.5.49 problem 49

Internal problem ID [6658]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 49
Date solved : Tuesday, September 30, 2025 at 03:50:35 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y+3 y^{\prime }+3 y^{\prime \prime }+y^{\prime \prime \prime }&=\left (-x^{2}+2\right ) {\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 32

ode:=y(x)+3*diff(y(x),x)+3*diff(diff(y(x),x),x)+diff(diff(diff(y(x),x),x),x) = (-x^2+2)/exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{-x} \left (x^{5}-60 c_3 \,x^{2}-20 x^{3}-60 c_2 x -60 c_1 \right )}{60} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 41

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

\begin{align*} y(x)&\to \frac {1}{60} e^{-x} \left (-x^5+20 x^3+60 c_3 x^2+60 c_2 x+60 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.216 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + x \left (C_{3} - \frac {x^{3}}{60} + \frac {x}{3}\right )\right )\right ) e^{- x} \]