56.4.14 problem 14

Internal problem ID [8903]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 14
Date solved : Monday, January 27, 2025 at 05:18:58 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*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 42

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=0);
 

\[ y = c_{1} x^{{3}/{2}} \left (1+\frac {3}{20} x +\frac {25}{224} x^{2}+\frac {1361}{17280} x^{3}+\frac {80753}{2365440} x^{4}+\frac {616517}{38707200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{2} x^{2}+\frac {2}{9} x^{3}+\frac {11}{120} x^{4}+\frac {82}{1575} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 80

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

\[ y(x)\to c_2 \left (\frac {82 x^5}{1575}+\frac {11 x^4}{120}+\frac {2 x^3}{9}+\frac {x^2}{2}+1\right )+c_1 \left (\frac {616517 x^5}{38707200}+\frac {80753 x^4}{2365440}+\frac {1361 x^3}{17280}+\frac {25 x^2}{224}+\frac {3 x}{20}+1\right ) x^{3/2} \]