Internal problem ID [4841]
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: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {16 x^{2} y^{\prime \prime }+16 x y^{\prime }+\left (16 x^{2}-1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 35
Order:=6; dsolve(16*x^2*diff(y(x),x$2)+16*x*diff(y(x),x)+(16*x^2-1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} \sqrt {x}\, \left (1-\frac {1}{5} x^{2}+\frac {1}{90} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{3} x^{2}+\frac {1}{42} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x^{\frac {1}{4}}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 52
AsymptoticDSolveValue[16*x^2*y''[x]+16*x*y'[x]+(16*x^2-1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [4]{x} \left (\frac {x^4}{90}-\frac {x^2}{5}+1\right )+\frac {c_2 \left (\frac {x^4}{42}-\frac {x^2}{3}+1\right )}{\sqrt [4]{x}} \]