Internal problem ID [6265]
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: 16.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \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: 32
Order:=8; dsolve((1+x^2)*diff(y(x),x$2)-2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (x^{2}+1\right ) y \relax (0)+\left (x +\frac {1}{3} x^{3}-\frac {1}{15} x^{5}+\frac {1}{35} x^{7}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 38
AsymptoticDSolveValue[(1+x^2)*y''[x]-2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (x^2+1\right )+c_2 \left (\frac {x^7}{35}-\frac {x^5}{15}+\frac {x^3}{3}+x\right ) \]