54.9.5 problem 5

Internal problem ID [8675]
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 : 5
Date solved : Monday, January 27, 2025 at 04:19:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+2 x \left (x^{2}+3\right ) y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 30

Order:=8; 
dsolve(x^2*(1+x^2)*diff(y(x),x$2)+2*x*(3+x^2)*diff(y(x),x)+6*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1-\frac {1}{3} x^{2}+\operatorname {O}\left (x^{8}\right )\right ) x +c_{2} \left (1-3 x^{2}+\operatorname {O}\left (x^{8}\right )\right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 26

AsymptoticDSolveValue[x^2*(1+x^2)*D[y[x],{x,2}]+2*x*(3+x^2)*D[y[x],x]+6*y[x]==0,y[x],{x,0,"8"-1}]
 

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