7.14.11 problem 11

Internal problem ID [2451]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 05:36:03 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 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: 0.037 (sec). Leaf size: 44

Order:=6; 
ode:=4*t*diff(diff(y(t),t),t)+3*diff(y(t),t)-3*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = c_1 \,t^{{1}/{4}} \left (1+\frac {3}{5} t +\frac {1}{10} t^{2}+\frac {1}{130} t^{3}+\frac {3}{8840} t^{4}+\frac {3}{309400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (1+t +\frac {3}{14} t^{2}+\frac {3}{154} t^{3}+\frac {3}{3080} t^{4}+\frac {9}{292600} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 81

ode=4*t*D[y[t],{t,2}]+3*D[y[t],t]-3*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 \sqrt [4]{t} \left (\frac {3 t^5}{309400}+\frac {3 t^4}{8840}+\frac {t^3}{130}+\frac {t^2}{10}+\frac {3 t}{5}+1\right )+c_2 \left (\frac {9 t^5}{292600}+\frac {3 t^4}{3080}+\frac {3 t^3}{154}+\frac {3 t^2}{14}+t+1\right ) \]

✓ Sympy. Time used: 0.286 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*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)
 

\[ y{\left (t \right )} = C_{2} \left (\frac {9 t^{5}}{292600} + \frac {3 t^{4}}{3080} + \frac {3 t^{3}}{154} + \frac {3 t^{2}}{14} + t + 1\right ) + C_{1} \sqrt [4]{t} \left (\frac {3 t^{4}}{8840} + \frac {t^{3}}{130} + \frac {t^{2}}{10} + \frac {3 t}{5} + 1\right ) + O\left (t^{6}\right ) \]