64.15.8 problem 8

Internal problem ID [13585]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 05:53:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+\left (2 x^{2}+\frac {5}{9}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 35

Order:=6; 
dsolve(x^2*diff(y(x),x$2)-x*diff(y(x),x)+(2*x^2+5/9)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{{1}/{3}} \left (1-\frac {3}{2} x^{2}+\frac {9}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{{5}/{3}} \left (1-\frac {3}{10} x^{2}+\frac {9}{320} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 52

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]-x*D[y[x],x]+(2*x^2+5/9)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {9 x^4}{32}-\frac {3 x^2}{2}+1\right ) \sqrt [3]{x}+c_1 \left (\frac {9 x^4}{320}-\frac {3 x^2}{10}+1\right ) x^{5/3} \]