64.21.3 problem 5

Internal problem ID [13585]
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 : 5
Date solved : Wednesday, March 05, 2025 at 10:04:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (t^{3}-2 t^{2}\right ) x^{\prime \prime }-\left (t^{3}+2 t^{2}-6 t \right ) x^{\prime }+\left (3 t^{2}-6\right ) x&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 15

ode:=(t^3-2*t^2)*diff(diff(x(t),t),t)-(t^3+2*t^2-6*t)*diff(x(t),t)+(3*t^2-6)*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.328 (sec). Leaf size: 115

ode=(t^3-2*t^2)*D[x[t],{t,2}]-(t^3+2*t^2-6*t)*D[x[t],t]+(3*t^2-6)*x[t]==0; 
ic={}; 
DSolve[{ode,ic},{x[t]},t,IncludeSingularSolutions->True]
 

\[ x(t)\to \exp \left (\int _1^t\frac {(K[1]-4) K[1]+2}{2 (K[1]-2) K[1]}dK[1]-\frac {1}{2} \int _1^t\left (-\frac {3}{K[2]}-1+\frac {1}{2-K[2]}\right )dK[2]\right ) \left (c_2 \int _1^t\exp \left (-2 \int _1^{K[3]}\frac {K[1]^2-4 K[1]+2}{2 (K[1]-2) K[1]}dK[1]\right )dK[3]+c_1\right ) \]

✓ Sympy. Time used: 0.931 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq((3*t**2 - 6)*x(t) + (t**3 - 2*t**2)*Derivative(x(t), (t, 2)) - (t**3 + 2*t**2 - 6*t)*Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

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