76.33.8 problem Ex. 8

Internal problem ID [20186]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. Examples on chapter VI, page 80
Problem number : Ex. 8
Date solved : Thursday, October 02, 2025 at 05:34:02 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }+y^{\prime \prime }&=x^{2} \left (b x +a \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 78

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

\[ y = -\frac {{\mathrm e}^{-\frac {x}{2}} \left (-c_2 \sqrt {3}+c_1 \right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {{\mathrm e}^{-\frac {x}{2}} \left (c_1 \sqrt {3}+c_2 \right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {b \,x^{5}}{20}+\frac {\left (a -3 b \right ) x^{4}}{12}-\frac {a \,x^{3}}{3}+3 x^{2} b +c_3 x +c_4 \]

✓ Mathematica. Time used: 0.617 (sec). Leaf size: 107

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

\begin{align*} y(x)&\to \frac {1}{12} \left (a (x-4) x^3+\frac {3}{5} b \left (x^3-5 x^2+60\right ) x^2+6 \left (\sqrt {3} c_1-c_2\right ) e^{-x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )-6 \left (c_1+\sqrt {3} c_2\right ) e^{-x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )\right )+c_4 x+c_3 \end{align*}

✓ Sympy. Time used: 0.123 (sec). Leaf size: 66

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

\[ y{\left (x \right )} = C_{1} + C_{2} x - \frac {a x^{3}}{3} + \frac {b x^{5}}{20} + 3 b x^{2} + \frac {x^{4} \left (a - 3 b\right )}{12} + \left (C_{3} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{4} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} \]