73.25.29 problem 35.5 (a)

Internal problem ID [15737]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.5 (a)
Date solved : Tuesday, January 28, 2025 at 08:06:18 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x -3\right ) y^{\prime \prime }+\left (x -3\right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 3 \end{align*}

✓ Solution by Maple

Time used: 0.067 (sec). Leaf size: 62

Order:=6; 
dsolve((x-3)*diff(y(x),x$2)+(x-3)*diff(y(x),x)+y(x)=0,y(x),type='series',x=3);
 

\[ y = c_{1} \left (x -3\right ) \left (1-\left (x -3\right )+\frac {1}{2} \left (x -3\right )^{2}-\frac {1}{6} \left (x -3\right )^{3}+\frac {1}{24} \left (x -3\right )^{4}-\frac {1}{120} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )+c_{2} \left (\ln \left (x -3\right ) \left (-\left (x -3\right )+\left (x -3\right )^{2}-\frac {1}{2} \left (x -3\right )^{3}+\frac {1}{6} \left (x -3\right )^{4}-\frac {1}{24} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )+\left (1-\left (x -3\right )+\frac {1}{4} \left (x -3\right )^{3}-\frac {5}{36} \left (x -3\right )^{4}+\frac {13}{288} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 105

AsymptoticDSolveValue[(x-3)*D[y[x],{x,2}]+(x-3)*D[y[x],x]+y[x]==0,y[x],{x,3,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {1}{24} (x-3)^5-\frac {1}{6} (x-3)^4+\frac {1}{2} (x-3)^3-(x-3)^2+x-3\right )+c_1 \left (\frac {1}{36} \left (-11 (x-3)^4+27 (x-3)^3-36 (x-3)^2+36\right )+\frac {1}{6} \left ((x-3)^3-3 (x-3)^2+6 (x-3)-6\right ) (x-3) \log (x-3)\right ) \]