73.25.6 problem 35.2 (f)

Internal problem ID [15714]
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 : 35.2 (f)
Date solved : Tuesday, January 28, 2025 at 08:05:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (x -5\right )^{2} y^{\prime \prime }+\left (x -5\right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 59

Order:=6; 
dsolve((x-5)^2*diff(y(x),x$2)+(x-5)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {2}{25} x^{2}-\frac {2}{125} x^{3}-\frac {7}{3750} x^{4}-\frac {1}{9375} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{10} x^{2}-\frac {1}{75} x^{3}-\frac {3}{500} x^{4}-\frac {1}{750} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 70

AsymptoticDSolveValue[(x-5)^2*D[y[x],{x,2}]+(x-5)*D[y[x],x]+4*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{9375}-\frac {7 x^4}{3750}-\frac {2 x^3}{125}-\frac {2 x^2}{25}+1\right )+c_2 \left (-\frac {x^5}{750}-\frac {3 x^4}{500}-\frac {x^3}{75}+\frac {x^2}{10}+x\right ) \]