62.25.2 problem Ex 2

Internal problem ID [12855]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 48. Page 103
Problem number : Ex 2
Date solved : Wednesday, March 05, 2025 at 08:48:27 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 30

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

\[ y = \frac {x^{2}}{3}+\frac {2 x}{9}+\frac {20}{27}+{\mathrm e}^{x} c_{1} +c_{2} {\mathrm e}^{-x}+c_3 \,{\mathrm e}^{3 x} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 42

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

\[ y(x)\to \frac {1}{27} \left (9 x^2+6 x+20\right )+c_1 e^{-x}+c_2 e^x+c_3 e^{3 x} \]

✓ Sympy. Time used: 0.204 (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)
 

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