49.15.6 problem 2

Internal problem ID [7692]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 130
Problem number : 2
Date solved : Monday, January 27, 2025 at 03:09:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 12

Order:=6; 
dsolve([diff(y(x),x$2)+(x-1)^2*diff(y(x),x)-(x-1)*y(x)=0,y(1) = 1, D(y)(1) = 0],y(x),type='series',x=1);
 

\[ y = 1+\frac {1}{6} \left (x -1\right )^{3}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 14

AsymptoticDSolveValue[{D[y[x],{x,2}]+(x-1)^2*D[y[x],x]-(x-1)*y[x]==0,{y[1]==1,Derivative[1][y][1]==0}},y[x],{x,1,"6"-1}]
 

\[ y(x)\to \frac {1}{6} (x-1)^3+1 \]