8.9.14 problem 17

Internal problem ID [2580]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2.2. Equal roots, reduction of order. Excercises page 149
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 05:47:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t y^{\prime \prime }-\left (1+3 t \right ) y^{\prime }+3 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{3 t} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 16

ode:=t*diff(diff(y(t),t),t)-(1+3*t)*diff(y(t),t)+3*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = c_2 \,{\mathrm e}^{3 t}+3 c_1 t +c_1 \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 25

ode=t*D[y[t],{t,2}]-(1+3*t)*D[y[t],t]+3*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to c_1 e^{3 t}-\frac {1}{9} c_2 (3 t+1) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), (t, 2)) - (3*t + 1)*Derivative(y(t), t) + 3*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

False