54.9.23 problem 24

Internal problem ID [8693]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 24
Date solved : Monday, January 27, 2025 at 04:19:23 PM
CAS classification : [[_Bessel, _modified]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+4\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 50

Order:=8; 
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)-(x^2+4)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} x^{4} \left (1+\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\frac {1}{23040} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (9 x^{4}+\frac {3}{4} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-144+36 x^{2}-\frac {1}{2} x^{6}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 74

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*D[y[x],x]-(x^2+4)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {11 x^6+36 x^4-576 x^2+2304}{2304 x^2}-\frac {1}{192} x^2 \left (x^2+12\right ) \log (x)\right )+c_2 \left (\frac {x^8}{23040}+\frac {x^6}{384}+\frac {x^4}{12}+x^2\right ) \]