13.9.12 problem 15

Internal problem ID [2398]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.2, Equal roots, reduction of order. Page 147
Problem number : 15
Date solved : Sunday, March 30, 2025 at 12:00:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 t +1\right ) y^{\prime \prime }-4 \left (t +1\right ) y^{\prime }+4 y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 15

ode:=(2*t+1)*diff(diff(y(t),t),t)-4*(t+1)*diff(y(t),t)+4*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = c_2 \,{\mathrm e}^{2 t}+c_1 t +c_1 \]

✓ Mathematica. Time used: 0.16 (sec). Leaf size: 23

ode=(2*t+1)*D[y[t],{t,2}]-4*(t+1)*D[y[t],t]+4*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to c_1 e^{2 t+1}-c_2 (t+1) \]

✓ Sympy. Time used: 1.088 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((2*t + 1)*Derivative(y(t), (t, 2)) - (4*t + 4)*Derivative(y(t), t) + 4*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - t^{4} r{\left (3 \right )} + \frac {17 t^{5} r{\left (3 \right )}}{10} + C_{2} \left (\frac {52 t^{5}}{15} - \frac {11 t^{4}}{3} - 4 t^{2} + 1\right ) + C_{1} t \left (- 2 t^{4} + 2 t^{3} + 2 t + 1\right ) + O\left (t^{6}\right ) \]