7.12.8 problem 8

Internal problem ID [2420]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8, Series solutions. Page 195
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 05:35:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 12

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 15

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

\[ y(t)\to \frac {1}{4} (t+1)^4+t+1 \]

✓ Sympy. Time used: 0.237 (sec). Leaf size: 29

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

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