Internal problem ID [4201]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XIV. page 177
Problem number: 16.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {\left (a^{2}+x^{2}\right ) y^{\prime \prime }+y^{\prime } x -y n^{2}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 72
Order:=6; dsolve((a^2+x^2)*diff(y(x),x$2)+x*diff(y(x),x)-n^2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1+\frac {n^{2} x^{2}}{2 a^{2}}+\frac {n^{2} \left (n^{2}-4\right ) x^{4}}{24 a^{4}}\right ) y \relax (0)+\left (x +\frac {\left (n^{2}-1\right ) x^{3}}{6 a^{2}}+\frac {\left (n^{4}-10 n^{2}+9\right ) x^{5}}{120 a^{4}}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 112
AsymptoticDSolveValue[(a^2+x^2)*y''[x]+x*y'[x]-n^2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {n^4 x^5}{120 a^4}-\frac {n^2 x^5}{12 a^4}+\frac {3 x^5}{40 a^4}+\frac {n^2 x^3}{6 a^2}-\frac {x^3}{6 a^2}+x\right )+c_1 \left (\frac {n^4 x^4}{24 a^4}-\frac {n^2 x^4}{6 a^4}+\frac {n^2 x^2}{2 a^2}+1\right ) \]