8.9.8 problem 11

Internal problem ID [2574]
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 : 11
Date solved : Tuesday, September 30, 2025 at 05:47:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-4 t y^{\prime }+\left (4 t^{2}-2\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 18

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

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

✗ Sympy

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

False