54.9.9 problem 9

Internal problem ID [8679]
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 : 9
Date solved : Monday, January 27, 2025 at 04:19:05 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (-x^{2}+1\right ) y^{\prime \prime }-\left (x^{2}+7\right ) y^{\prime }+4 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 32

Order:=8; 
dsolve(x*(1-x^2)*diff(y(x),x$2)-(7+x^2)*diff(y(x),x)+4*x*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{8} \left (1+3 x^{2}+6 x^{4}+10 x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-203212800-67737600 x^{2}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 38

AsymptoticDSolveValue[x*(1-x^2)*D[y[x],{x,2}]-(7+x^2)*D[y[x],x]+4*x*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^2}{3}+1\right )+c_2 \left (10 x^{14}+6 x^{12}+3 x^{10}+x^8\right ) \]