14.12.5 problem 5

Internal problem ID [2615]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8. Series solutions. Excercises page 197
Problem number : 5
Date solved : Monday, January 27, 2025 at 06:04:37 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*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 14

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

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 19

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

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