Internal problem ID [5848]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT
SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number: 18.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} y^{\prime \prime }-x y^{\prime }+y \left (x^{2}+1\right )=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 37
Order:=8; dsolve(2*x^2*diff(y(x),x$2)-x*diff(y(x),x)+(x^2+1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{6} x^{2}+\frac {1}{168} x^{4}-\frac {1}{11088} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x \left (1-\frac {1}{10} x^{2}+\frac {1}{360} x^{4}-\frac {1}{28080} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 62
AsymptoticDSolveValue[2*x^2*y''[x]-x*y'[x]+(x^2+1)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 x \left (-\frac {x^6}{28080}+\frac {x^4}{360}-\frac {x^2}{10}+1\right )+c_2 \sqrt {x} \left (-\frac {x^6}{11088}+\frac {x^4}{168}-\frac {x^2}{6}+1\right ) \]