23.5.2 problem 3(a)

Internal problem ID [4179]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(a)
Date solved : Monday, January 27, 2025 at 08:41:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {\left (-3 x^{2}+x \right ) y^{\prime }}{2 x^{3}+2 x^{2}}+\frac {y}{2 x^{3}+2 x^{2}}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 40

Order:=6; 
dsolve(diff(y(x),x$2)-(x-3*x^2)/(2*(x^2+x^3))*diff(y(x),x)+y(x)/(2*(x^2+x^3))=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 59

AsymptoticDSolveValue[D[y[x],{x,2}]-(x-3*x^2)/(2*(x^2+x^3))*D[y[x],x]+y[x]/(2*(x^2+x^3))==0,{y[x]},{x,0,"6"-1}]
 

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