23.5.44 problem 44

Internal problem ID [6653]
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 : 44
Date solved : Tuesday, September 30, 2025 at 03:50:33 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} 3 y-y^{\prime }-3 y^{\prime \prime }+y^{\prime \prime \prime }&=x^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 30

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

\[ y = \frac {x^{2}}{3}+\frac {2 x}{9}+\frac {20}{27}+c_1 \,{\mathrm e}^{x}+c_2 \,{\mathrm e}^{-x}+c_3 \,{\mathrm e}^{3 x} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 42

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

\begin{align*} y(x)&\to \frac {1}{27} \left (9 x^2+6 x+20\right )+c_1 e^{-x}+c_2 e^x+c_3 e^{3 x} \end{align*}

✓ Sympy. Time used: 0.116 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{1} e^{- x} + C_{2} e^{x} + C_{3} e^{3 x} + \frac {x^{2}}{3} + \frac {2 x}{9} + \frac {20}{27} \]