7.14.19 problem 19

Internal problem ID [444]
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 : 19
Date solved : Monday, January 27, 2025 at 02:53:41 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Solution by Maple

Time used: 0.000 (sec). Leaf size: 14

Order:=6; 
dsolve([(2*x-x^2)*diff(y(x),x$2)-6*(x-1)*diff(y(x),x)-4*y(x)=0,y(1) = 0, D(y)(1) = 1],y(x),type='series',x=1);
 

\[ y = \left (x -1\right )+\frac {5}{3} \left (x -1\right )^{3}+\frac {7}{3} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 24

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

\[ y(x)\to \frac {7}{3} (x-1)^5+\frac {5}{3} (x-1)^3+x-1 \]