23.5.12 problem 3(k)

Internal problem ID [4189]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(k)
Date solved : Monday, January 27, 2025 at 08:41:34 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 44

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

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

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 47

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

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