52.2.22 problem 22

Internal problem ID [8273]
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
Date solved : Monday, January 27, 2025 at 03:47:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {4}{9}\right ) y&=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:=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);
 

\[ y = \frac {c_{2} x^{{4}/{3}} \left (1-\frac {3}{20} x^{2}+\frac {9}{1280} x^{4}-\frac {9}{56320} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} \left (1-\frac {3}{4} x^{2}+\frac {9}{128} x^{4}-\frac {9}{3584} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x^{{2}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 66

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-4/9)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 x^{2/3} \left (-\frac {9 x^6}{56320}+\frac {9 x^4}{1280}-\frac {3 x^2}{20}+1\right )+\frac {c_2 \left (-\frac {9 x^6}{3584}+\frac {9 x^4}{128}-\frac {3 x^2}{4}+1\right )}{x^{2/3}} \]