Internal problem ID [5852]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT
SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number: 22.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {4}{9}\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 39
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-4/9)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {3}{20} x^{2}+\frac {9}{1280} x^{4}-\frac {9}{56320} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+c_{1} \left (1-\frac {3}{4} x^{2}+\frac {9}{128} x^{4}-\frac {9}{3584} x^{6}+\mathrm {O}\left (x^{8}\right )\right )}{x^{\frac {2}{3}}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 66
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x^2-4/9)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 x^{2/3} \left (-\frac {9 x^6}{56320}+\frac {9 x^4}{1280}-\frac {3 x^2}{20}+1\right )+\frac {c_2 \left (-\frac {9 x^6}{3584}+\frac {9 x^4}{128}-\frac {3 x^2}{4}+1\right )}{x^{2/3}} \]