64.21.2 problem 1 (b)
Internal
problem
ID
[13592]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
11,
The
nth
order
homogeneous
linear
differential
equation.
Section
11.6,
Exercises
page
567
Problem
number
:
1
(b)
Date
solved
:
Monday, March 31, 2025 at 08:02:05 AM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} t^{3} x^{\prime \prime \prime }-3 t^{2} x^{\prime \prime }+6 t x^{\prime }-6 x&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 16
ode:=t^3*diff(diff(diff(x(t),t),t),t)-3*t^2*diff(diff(x(t),t),t)+6*t*diff(x(t),t)-6*x(t) = 0;
dsolve(ode,x(t), singsol=all);
\[
x = t \left (c_1 \,t^{2}+c_2 t +c_3 \right )
\]
✓ Mathematica. Time used: 0.03 (sec). Leaf size: 507
ode=t^3*D[x[t],{t,3}]-3*t^2*D[x[t],{t,2}]+6*D[x[t],t]-6*x[t]==0;
ic={};
DSolve[{ode,ic},{x[t]},t,IncludeSingularSolutions->True]
\[
x(t)\to c_3 6^{-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,3\right ]} t^{\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,3\right ]} \, _1F_2\left (-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,3\right ];1+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,3\right ],1+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,2\right ]-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,3\right ];-\frac {6}{t}\right )+c_2 6^{-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,2\right ]} t^{\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,2\right ]} \, _1F_2\left (-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,2\right ];1+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,1\right ]-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,2\right ],1-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,2\right ]+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,3\right ];-\frac {6}{t}\right )+c_1 6^{-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,1\right ]} t^{\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,1\right ]} \, _1F_2\left (-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,1\right ];1-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,1\right ]+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,2\right ],1-\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,1\right ]+\text {Root}\left [\text {$\#$1}^3-6 \text {$\#$1}^2+5 \text {$\#$1}-6\&,3\right ];-\frac {6}{t}\right )
\]
✓ Sympy. Time used: 0.196 (sec). Leaf size: 14
from sympy import *
t = symbols("t")
x = Function("x")
ode = Eq(t**3*Derivative(x(t), (t, 3)) - 3*t**2*Derivative(x(t), (t, 2)) + 6*t*Derivative(x(t), t) - 6*x(t),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
\[
x{\left (t \right )} = t \left (C_{1} + C_{2} t + C_{3} t^{2}\right )
\]