Internal problem ID [5272]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 3. Linear equations with variable coefficients. Page 130
Problem number: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y=0} \end {gather*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = 1] \end {align*}
With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 14
Order:=6; dsolve([(1+x^2)*diff(y(x),x$2)+y(x)=0,y(0) = 0, D(y)(0) = 1],y(x),type='series',x=0);
\[ y \relax (x ) = x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}+\mathrm {O}\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 19
AsymptoticDSolveValue[{(1+x^2)*y''[x]+y[x]==0,{y[0]==0,y'[0]==1}},y[x],{x,0,5}]
\[ y(x)\to \frac {7 x^5}{120}-\frac {x^3}{6}+x \]