23.5.6 problem 3(e)

Internal problem ID [4183]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(e)
Date solved : Monday, January 27, 2025 at 08:41:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {\left (x^{2}+4 x +2\right ) \left (\left (1-x \right ) y^{\prime }+y\right )}{x \left (-x^{2}+2\right )}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 44

Order:=6; 
dsolve(diff(y(x),x$2)-(x^2+4*x+2)/(x*(2-x^2))*( (1-x)*diff(y(x),x)+y(x) )=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 65

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

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