70.12.17 problem 29

Internal problem ID [18860]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 29
Date solved : Thursday, October 02, 2025 at 03:32:18 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} t^{2} y^{\prime \prime }-4 t y^{\prime }+6 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=t^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 13

ode:=t^2*diff(diff(y(t),t),t)-4*t*diff(y(t),t)+6*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = t^{2} \left (c_2 t +c_1 \right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 16

ode=t^2*D[y[t],{t,2}]-4*t*D[y[t],t]+6*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to t^2 (c_2 t+c_1) \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) - 4*t*Derivative(y(t), t) + 6*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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