14.15.1 problem 1

Internal problem ID [2666]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8.3, Equal roots and roots differing by an integer. Excercises page 223
Problem number : 1
Date solved : Sunday, March 30, 2025 at 12:13:03 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 44

Order:=6; 
ode:=t*diff(diff(y(t),t),t)+diff(y(t),t)-4*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \left (c_2 \ln \left (t \right )+c_1 \right ) \left (1+4 t +4 t^{2}+\frac {16}{9} t^{3}+\frac {4}{9} t^{4}+\frac {16}{225} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (\left (-8\right ) t -12 t^{2}-\frac {176}{27} t^{3}-\frac {50}{27} t^{4}-\frac {1096}{3375} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 105

ode=t*D[y[t],{t,2}]+D[y[t],t]-4*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 \left (\frac {16 t^5}{225}+\frac {4 t^4}{9}+\frac {16 t^3}{9}+4 t^2+4 t+1\right )+c_2 \left (-\frac {1096 t^5}{3375}-\frac {50 t^4}{27}-\frac {176 t^3}{27}-12 t^2+\left (\frac {16 t^5}{225}+\frac {4 t^4}{9}+\frac {16 t^3}{9}+4 t^2+4 t+1\right ) \log (t)-8 t\right ) \]

✓ Sympy. Time used: 0.723 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), (t, 2)) - 4*y(t) + Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{1} \left (\frac {16 t^{5}}{225} + \frac {4 t^{4}}{9} + \frac {16 t^{3}}{9} + 4 t^{2} + 4 t + 1\right ) + O\left (t^{6}\right ) \]