54.3.6 problem 6

Internal problem ID [8562]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 6
Date solved : Monday, January 27, 2025 at 04:16:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }-4 x y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 25

Order:=8; 
dsolve((1+x^2)*diff(y(x),x$2)-4*x*diff(y(x),x)+6*y(x)=0,y(x),type='series',x=0);
 

\[ y = y \left (0\right )+y^{\prime }\left (0\right ) x -3 y \left (0\right ) x^{2}-\frac {y^{\prime }\left (0\right ) x^{3}}{3} \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 26

AsymptoticDSolveValue[(1+x^2)*D[y[x],{x,2}]-4*x*D[y[x],x]+6*y[x]==0,y[x],{x,0,"8"-1}]
 

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