Internal problem ID [6153]
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: 18.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+2 x y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 44
Order:=8; dsolve(diff(y(x),x$2)+2*x*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-x^{2}+\frac {1}{2} x^{4}-\frac {1}{6} x^{6}\right ) y \relax (0)+\left (x -\frac {2}{3} x^{3}+\frac {4}{15} x^{5}-\frac {8}{105} x^{7}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 54
AsymptoticDSolveValue[y''[x]+2*x*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {8 x^7}{105}+\frac {4 x^5}{15}-\frac {2 x^3}{3}+x\right )+c_1 \left (-\frac {x^6}{6}+\frac {x^4}{2}-x^2+1\right ) \]