64.15.7 problem 7

Internal problem ID [13584]
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 : 7
Date solved : Tuesday, January 28, 2025 at 05:53:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+\left (x^{2}+\frac {8}{9}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 35

Order:=6; 
dsolve(x^2*diff(y(x),x$2)-x*diff(y(x),x)+(x^2+8/9)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{{2}/{3}} \left (1-\frac {3}{8} x^{2}+\frac {9}{320} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{{4}/{3}} \left (1-\frac {3}{16} x^{2}+\frac {9}{896} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 52

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]-x*D[y[x],x]+(x^2+8/9)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {9 x^4}{896}-\frac {3 x^2}{16}+1\right ) x^{4/3}+c_2 \left (\frac {9 x^4}{320}-\frac {3 x^2}{8}+1\right ) x^{2/3} \]