40.2.18 problem 20

Internal problem ID [8607]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 05:39:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 t \left (1+t \right ) y^{\prime \prime }+t y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 42

Order:=6; 
ode:=3*t*(t+1)*diff(diff(y(t),t),t)+t*diff(y(t),t)-y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = c_1 t \left (1+\operatorname {O}\left (t^{6}\right )\right )+\ln \left (t \right ) \left (\frac {1}{3} t +\operatorname {O}\left (t^{6}\right )\right ) c_2 +\left (1-\frac {1}{3} t -\frac {2}{9} t^{2}+\frac {7}{81} t^{3}-\frac {35}{729} t^{4}+\frac {91}{2916} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 43

ode=3*t*(1+t)*D[y[t],{t,2}]+t*D[y[t],t]-y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

\[ y(t)\to c_1 \left (\frac {1}{729} \left (-35 t^4+63 t^3-162 t^2+243 t+729\right )+\frac {1}{3} t \log (t)\right )+c_2 t \]

✓ Sympy. Time used: 0.309 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(3*t*(t + 1)*Derivative(y(t), (t, 2)) + t*Derivative(y(t), t) - y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{1} t \left (\frac {t^{4}}{2880} + \frac {t^{3}}{144} + \frac {t^{2}}{12} + \frac {t}{2} + 1\right ) + O\left (t^{6}\right ) \]