54.9.12 problem 12

Internal problem ID [8682]
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 : 12
Date solved : Monday, January 27, 2025 at 04:19:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 38

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

\[ y = x \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {3}{32} x^{4}-\frac {11}{576} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 86

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

\[ y(x)\to c_1 x \left (\frac {x^6}{48}+\frac {x^4}{8}+\frac {x^2}{2}+1\right )+c_2 \left (x \left (-\frac {11 x^6}{576}-\frac {3 x^4}{32}-\frac {x^2}{4}\right )+x \left (\frac {x^6}{48}+\frac {x^4}{8}+\frac {x^2}{2}+1\right ) \log (x)\right ) \]