56.4.15 problem 15

Internal problem ID [8904]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 15
Date solved : Monday, January 27, 2025 at 05:18:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 46

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

\[ y = c_{1} \sqrt {x -2}\, \left (1-\frac {23}{12} \left (x -2\right )+\frac {127}{160} \left (x -2\right )^{2}+\frac {1621}{40320} \left (x -2\right )^{3}-\frac {426599}{5806080} \left (x -2\right )^{4}+\frac {4670443}{425779200} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right )\right )+c_{2} \left (1-6 \left (x -2\right )+\frac {31}{6} \left (x -2\right )^{2}-\frac {37}{45} \left (x -2\right )^{3}-\frac {299}{840} \left (x -2\right )^{4}+\frac {6743}{56700} \left (x -2\right )^{5}+\operatorname {O}\left (\left (x -2\right )^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 105

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

\[ y(x)\to c_1 \left (\frac {4670443 (x-2)^5}{425779200}-\frac {426599 (x-2)^4}{5806080}+\frac {1621 (x-2)^3}{40320}+\frac {127}{160} (x-2)^2-\frac {23 (x-2)}{12}+1\right ) \sqrt {x-2}+c_2 \left (\frac {6743 (x-2)^5}{56700}-\frac {299}{840} (x-2)^4-\frac {37}{45} (x-2)^3+\frac {31}{6} (x-2)^2-6 (x-2)+1\right ) \]