26.11 problem 36.2 (k)

Internal problem ID [14015]

Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page 739
Problem number: 36.2 (k).
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]

\[ \boxed {\left (x -3\right ) y^{\prime \prime }+\left (x -3\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 3\).

Solution by Maple

Time used: 0.047 (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);
 

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

Solution by Mathematica

Time used: 0.02 (sec). Leaf size: 105

AsymptoticDSolveValue[(x-3)*y''[x]+(x-3)*y'[x]+y[x]==0,y[x],{x,3,5}]
 

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