89.15.14 problem 14

Internal problem ID [24648]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Oral Exercises at page 131
Problem number : 14
Date solved : Thursday, October 02, 2025 at 10:46:42 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-5 y^{\prime }&=12 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 81

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-5*diff(y(x),x) = 12; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{-\frac {5^{{1}/{3}} x}{2}} 5^{{2}/{3}} \left (c_3 \sqrt {3}+c_2 \right ) \cos \left (\frac {\sqrt {3}\, 5^{{1}/{3}} x}{2}\right )}{10}+\frac {{\mathrm e}^{-\frac {5^{{1}/{3}} x}{2}} 5^{{2}/{3}} \left (\sqrt {3}\, c_2 -c_3 \right ) \sin \left (\frac {\sqrt {3}\, 5^{{1}/{3}} x}{2}\right )}{10}+\frac {c_1 \,{\mathrm e}^{5^{{1}/{3}} x} 5^{{2}/{3}}}{5}-\frac {12 x}{5}+c_4 \]

✓ Mathematica. Time used: 0.152 (sec). Leaf size: 129

ode=D[y[x],{x,4}]-5*D[y[x],{x,1}]==12; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{10} e^{-\frac {1}{2} \sqrt [3]{5} x} \left (2 e^{\frac {\sqrt [3]{5} x}{2}} \left (-12 x+5^{2/3} c_3 e^{\sqrt [3]{5} x}\right )-5^{2/3} \left (c_1+\sqrt {3} c_2\right ) \cos \left (\frac {1}{2} \sqrt {3} \sqrt [3]{5} x\right )+5^{2/3} \left (\sqrt {3} c_1-c_2\right ) \sin \left (\frac {1}{2} \sqrt {3} \sqrt [3]{5} x\right )\right )+c_4 \end{align*}

✓ Sympy. Time used: 0.174 (sec). Leaf size: 63

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-5*Derivative(y(x), x) + Derivative(y(x), (x, 4)) - 12,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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