15.10.21 problem 21

Internal problem ID [3108]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 18, page 82
Problem number : 21
Date solved : Sunday, March 30, 2025 at 01:18:37 AM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+4 y^{\prime \prime }-12 y^{\prime }+16 y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 41

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-3*diff(diff(diff(y(x),x),x),x)+4*diff(diff(y(x),x),x)-12*diff(y(x),x)+16*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_4 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {15}\, x}{2}\right )+c_3 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {15}\, x}{2}\right )+{\mathrm e}^{2 x} \left (c_2 x +c_1 \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 57

ode=D[y[x],{x,4}]-3*D[y[x],{x,3}]+4*D[y[x],{x,2}]-12*D[y[x],x]+16*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-x/2} \left (e^{5 x/2} (c_4 x+c_3)+c_2 \cos \left (\frac {\sqrt {15} x}{2}\right )+c_1 \sin \left (\frac {\sqrt {15} x}{2}\right )\right ) \]

✓ Sympy. Time used: 0.247 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(16*y(x) - 12*Derivative(y(x), x) + 4*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} x\right ) e^{2 x} + \left (C_{3} \sin {\left (\frac {\sqrt {15} x}{2} \right )} + C_{4} \cos {\left (\frac {\sqrt {15} x}{2} \right )}\right ) e^{- \frac {x}{2}} \]