54.9.19 problem 20

Internal problem ID [8689]
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 : 20
Date solved : Monday, January 27, 2025 at 04:19:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 32

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

\[ y = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{2} x^{2}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {3}{4} x^{2}-\frac {1}{32} x^{4}-\frac {1}{576} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 53

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

\[ y(x)\to c_1 \left (1-\frac {x^2}{2}\right )+c_2 \left (-\frac {x^6}{576}-\frac {x^4}{32}+\frac {3 x^2}{4}+\left (1-\frac {x^2}{2}\right ) \log (x)\right ) \]