7.14.18 problem 18

Internal problem ID [443]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.2 (Series solution near ordinary points). Problems at page 216
Problem number : 18
Date solved : Monday, January 27, 2025 at 02:53:41 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 21

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

\[ y(x)\to \frac {1}{4} (x-1)^4-(x-1)^2+2 \]