7.15.4 problem 4

Internal problem ID [2471]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8.3, The method of Frobenius. Equal roots, and roots differering by an integer. Page 223
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 05:36:21 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t y^{\prime \prime }+3 y^{\prime }-3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 62

Order:=6; 
ode:=t*diff(diff(y(t),t),t)+3*diff(y(t),t)-3*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \frac {c_1 \left (1+t +\frac {3}{8} t^{2}+\frac {3}{40} t^{3}+\frac {3}{320} t^{4}+\frac {9}{11200} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) t^{2}+c_2 \left (\ln \left (t \right ) \left (9 t^{2}+9 t^{3}+\frac {27}{8} t^{4}+\frac {27}{40} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (-2+6 t -12 t^{3}-\frac {225}{32} t^{4}-\frac {1413}{800} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right )}{t^{2}} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 78

ode=t*D[y[t],{t,2}]+3*D[y[t],t]-3*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_2 \left (\frac {3 t^4}{320}+\frac {3 t^3}{40}+\frac {3 t^2}{8}+t+1\right )+c_1 \left (\frac {279 t^4+528 t^3+144 t^2-192 t+64}{64 t^2}-\frac {9}{16} \left (3 t^2+8 t+8\right ) \log (t)\right ) \]

✓ Sympy. Time used: 0.220 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), (t, 2)) - 3*y(t) + 3*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{1} \left (\frac {9 t^{5}}{11200} + \frac {3 t^{4}}{320} + \frac {3 t^{3}}{40} + \frac {3 t^{2}}{8} + t + 1\right ) + O\left (t^{6}\right ) \]