63.12.8 problem 3

Internal problem ID [13091]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.2 Variation of parameters. Exercises page 124
Problem number : 3
Date solved : Wednesday, March 05, 2025 at 09:17:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} x^{\prime \prime }-3 t x^{\prime }+3 x&=4 t^{7} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 20

ode:=t^2*diff(diff(x(t),t),t)-3*t*diff(x(t),t)+3*x(t) = 4*t^7; 
dsolve(ode,x(t), singsol=all);
 

\[ x \left (t \right ) = \frac {\left (t^{6}+3 c_{1} t^{2}+6 c_{2} \right ) t}{6} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 23

ode=t^2*D[x[t],{t,2}]-3*t*D[x[t],t]+3*x[t]==4*t^7; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to \frac {t^7}{6}+c_2 t^3+c_1 t \]

✓ Sympy. Time used: 0.346 (sec). Leaf size: 15

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-4*t**7 + t**2*Derivative(x(t), (t, 2)) - 3*t*Derivative(x(t), t) + 3*x(t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = t \left (C_{1} + C_{2} t^{2} + \frac {t^{6}}{6}\right ) \]