45.3.1 problem 1

Internal problem ID [7259]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.3.1 page 250
Problem number : 1
Date solved : Monday, January 27, 2025 at 02:49:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{9}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 36

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-1/9)*y(x)=0,y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 52

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

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