23.5.58 problem 58

Internal problem ID [6667]
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 : 58
Date solved : Tuesday, September 30, 2025 at 03:50:39 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 24

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

\[ y = {\mathrm e}^{a x} \left (\frac {1}{6} x^{3}+c_1 +c_2 x +c_3 \,x^{2}\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 34

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

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

✓ Sympy. Time used: 0.185 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + x \left (C_{3} + \frac {x}{6}\right )\right )\right ) e^{a x} \]