46.2.16 problem 18

Internal problem ID [7319]
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 : 18
Date solved : Wednesday, March 05, 2025 at 04:23:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 \left (t^{2}-3 t +2\right ) y^{\prime \prime }-2 y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 59

Order:=6; 
ode:=4*(t^2-3*t+2)*diff(diff(y(t),t),t)-2*diff(y(t),t)+y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \left (1-\frac {1}{16} t^{2}-\frac {7}{192} t^{3}-\frac {73}{3072} t^{4}-\frac {1037}{61440} t^{5}\right ) y \left (0\right )+\left (t +\frac {1}{8} t^{2}+\frac {5}{96} t^{3}+\frac {47}{1536} t^{4}+\frac {643}{30720} t^{5}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 70

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

\[ y(t)\to c_1 \left (-\frac {1037 t^5}{61440}-\frac {73 t^4}{3072}-\frac {7 t^3}{192}-\frac {t^2}{16}+1\right )+c_2 \left (\frac {643 t^5}{30720}+\frac {47 t^4}{1536}+\frac {5 t^3}{96}+\frac {t^2}{8}+t\right ) \]

✓ Sympy. Time used: 1.047 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((4*t**2 - 12*t + 8)*Derivative(y(t), (t, 2)) + y(t) - 2*Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (t \right )} = C_{2} \left (- \frac {73 t^{4}}{3072} - \frac {7 t^{3}}{192} - \frac {t^{2}}{16} + 1\right ) + C_{1} t \left (\frac {47 t^{3}}{1536} + \frac {5 t^{2}}{96} + \frac {t}{8} + 1\right ) + O\left (t^{6}\right ) \]