13.9.8 problem 11

Internal problem ID [2394]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.2, Equal roots, reduction of order. Page 147
Problem number : 11
Date solved : Sunday, March 30, 2025 at 12:00:12 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*}

✓ 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: 0.024 (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]
 

\[ y(t)\to e^{t^2} (c_2 t+c_1) \]

✓ Sympy. Time used: 0.926 (sec). Leaf size: 37

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)
 

\[ y{\left (t \right )} = \frac {7 t^{5} r{\left (3 \right )}}{10} + C_{2} \left (\frac {t^{4}}{2} + t^{2} + 1\right ) + C_{1} t \left (1 - \frac {t^{4}}{5}\right ) + O\left (t^{6}\right ) \]