52.2.20 problem 20

Internal problem ID [8271]
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 : 20
Date solved : Monday, January 27, 2025 at 03:47:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 55

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

\[ y = c_{1} x^{{1}/{3}} \left (1+\frac {3}{2} x +\frac {9}{20} x^{2}+\frac {9}{160} x^{3}+\frac {27}{7040} x^{4}+\frac {81}{492800} x^{5}+\frac {81}{16755200} x^{6}+\frac {243}{2345728000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{{2}/{3}} \left (1+\frac {3}{4} x +\frac {9}{56} x^{2}+\frac {9}{560} x^{3}+\frac {27}{29120} x^{4}+\frac {81}{2329600} x^{5}+\frac {81}{88524800} x^{6}+\frac {243}{13632819200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 118

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

\[ y(x)\to c_2 \sqrt [3]{x} \left (\frac {243 x^7}{2345728000}+\frac {81 x^6}{16755200}+\frac {81 x^5}{492800}+\frac {27 x^4}{7040}+\frac {9 x^3}{160}+\frac {9 x^2}{20}+\frac {3 x}{2}+1\right )+c_1 x^{2/3} \left (\frac {243 x^7}{13632819200}+\frac {81 x^6}{88524800}+\frac {81 x^5}{2329600}+\frac {27 x^4}{29120}+\frac {9 x^3}{560}+\frac {9 x^2}{56}+\frac {3 x}{4}+1\right ) \]