Internal problem ID [4838]
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.
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 {1}{9}\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 35
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);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {2}{3}} \left (1-\frac {3}{16} x^{2}+\frac {9}{896} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {3}{8} x^{2}+\frac {9}{320} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 52
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x^2-1/9)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {9 x^4}{896}-\frac {3 x^2}{16}+1\right )+\frac {c_2 \left (\frac {9 x^4}{320}-\frac {3 x^2}{8}+1\right )}{\sqrt [3]{x}} \]