Internal problem ID [6146]
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: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {\left (x^{2}+4\right ) y^{\prime \prime }+6 x y^{\prime }+4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 44
Order:=8; dsolve((x^2+4)*diff(y(x),x$2)+6*x*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {3}{16} x^{4}-\frac {1}{16} x^{6}\right ) y \relax (0)+\left (x -\frac {5}{12} x^{3}+\frac {7}{48} x^{5}-\frac {3}{64} x^{7}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 56
AsymptoticDSolveValue[(x^2+4)*y''[x]+6*x*y'[x]+4*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {3 x^7}{64}+\frac {7 x^5}{48}-\frac {5 x^3}{12}+x\right )+c_1 \left (-\frac {x^6}{16}+\frac {3 x^4}{16}-\frac {x^2}{2}+1\right ) \]