75.5.2 problem 2

Internal problem ID [19950]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter IV. Ordinary linear differential equations with constant coefficients. Exercises at page 58
Problem number : 2
Date solved : Thursday, October 02, 2025 at 05:04:24 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+4 y^{\prime \prime }+3 y^{\prime }&=x^{2} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

ode:=diff(diff(diff(y(x),x),x),x)+4*diff(diff(y(x),x),x)+3*diff(y(x),x) = x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{3}}{9}-\frac {4 x^{2}}{9}-\frac {{\mathrm e}^{-3 x} c_2}{3}-{\mathrm e}^{-x} c_1 +\frac {26 x}{27}+c_3 \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 47

ode=D[y[x],{x,3}]+4*D[y[x],{x,2}]+3*D[y[x],x]==x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^3}{9}-\frac {4 x^2}{9}+\frac {26 x}{27}-\frac {1}{3} c_1 e^{-3 x}-c_2 e^{-x}+c_3 \end{align*}

✓ Sympy. Time used: 0.133 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 + 3*Derivative(y(x), x) + 4*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{2} e^{- 3 x} + C_{3} e^{- x} + \frac {x^{3}}{9} - \frac {4 x^{2}}{9} + \frac {26 x}{27} \]