7.12.5 problem 5

Internal problem ID [2417]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8, Series solutions. Page 195
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 05:35:35 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} t \left (2-t \right ) y^{\prime \prime }-6 \left (t -1\right ) y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 14

Order:=6; 
ode:=t*(2-t)*diff(diff(y(t),t),t)-6*(t-1)*diff(y(t),t)-4*y(t) = 0; 
ic:=[y(1) = 1, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(t),type='series',t=1);
 

\[ y = 1+2 \left (t -1\right )^{2}+3 \left (t -1\right )^{4}+\operatorname {O}\left (\left (t -1\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 19

ode=t*(2-t)*D[y[t],{t,2}]-6*(t-1)*D[y[t],t]-4*y[t]==0; 
ic={y[1]==1,Derivative[1][y][1]==0}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,1,5}]
 

\[ y(t)\to 3 (t-1)^4+2 (t-1)^2+1 \]

✓ Sympy. Time used: 0.295 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*(2 - t)*Derivative(y(t), (t, 2)) - (6*t - 6)*Derivative(y(t), t) - 4*y(t),0) 
ics = {y(1): 1, Subs(Derivative(y(t), t), t, 1): 0} 
dsolve(ode,func=y(t),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

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