3.2 problem 2

Internal problem ID [4839]

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: 2.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [_Bessel]

Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (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: 0.024 (sec). Leaf size: 47

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

\[ y \relax (x ) = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (x^{2}-\frac {1}{8} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x} \]

Solution by Mathematica

Time used: 0.01 (sec). Leaf size: 58

AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x^2-1)*y[x]==0,y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{192}-\frac {x^3}{8}+x\right )+c_1 \left (\frac {1}{16} x \left (x^2-8\right ) \log (x)-\frac {5 x^4-16 x^2-64}{64 x}\right ) \]