Internal problem ID [14820]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number: 19 (b).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Bessel]
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (-k^{2}+x^{2}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 77
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-k^2)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{-k} \left (1+\frac {1}{4 k -4} x^{2}+\frac {1}{32} \frac {1}{\left (-2+k \right ) \left (-1+k \right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{k} \left (1-\frac {1}{4 k +4} x^{2}+\frac {1}{32} \frac {1}{\left (k +2\right ) \left (1+k \right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 160
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x^2-k^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^4}{\left (-k^2-k+(1-k) (2-k)+2\right ) \left (-k^2-k+(3-k) (4-k)+4\right )}-\frac {x^2}{-k^2-k+(1-k) (2-k)+2}+1\right ) x^{-k}+c_1 \left (\frac {x^4}{\left (-k^2+k+(k+1) (k+2)+2\right ) \left (-k^2+k+(k+3) (k+4)+4\right )}-\frac {x^2}{-k^2+k+(k+1) (k+2)+2}+1\right ) x^k \]