74.17.19 problem 19 (b)

Internal problem ID [16550]
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)
Date solved : Tuesday, January 28, 2025 at 09:10:38 AM
CAS classification : [_Bessel]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 73

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);
 

\[ y = c_{1} x^{-k} \left (1+\frac {1}{4 k -4} x^{2}+\frac {1}{32} \frac {1}{\left (-2+k \right ) \left (k -1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{k} \left (1-\frac {1}{4 k +4} x^{2}+\frac {1}{32} \frac {1}{\left (k +2\right ) \left (k +1\right )} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 160

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

\[ y(x)\to c_2 \left (\frac {x^4}{\left (-k^2-k+(1-k) (2-k)+2\right ) \left (-k^2-k+(3-k) (4-k)+4\right )}-\frac {x^2}{-k^2-k+(1-k) (2-k)+2}+1\right ) x^{-k}+c_1 \left (\frac {x^4}{\left (-k^2+k+(k+1) (k+2)+2\right ) \left (-k^2+k+(k+3) (k+4)+4\right )}-\frac {x^2}{-k^2+k+(k+1) (k+2)+2}+1\right ) x^k \]