34.3.14 problem 16

Internal problem ID [6053]
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
Date solved : Monday, January 27, 2025 at 01:34:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (a^{2}+x^{2}\right ) y^{\prime \prime }+x y^{\prime }-n^{2} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 77

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);
 

\[ y = \left (1+\frac {n^{2} x^{2}}{2 a^{2}}+\frac {n^{2} \left (n^{2}-4\right ) x^{4}}{24 a^{4}}\right ) y \left (0\right )+\left (x +\frac {\left (n^{2}-1\right ) x^{3}}{6 a^{2}}+\frac {\left (n^{4}-10 n^{2}+9\right ) x^{5}}{120 a^{4}}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 112

AsymptoticDSolveValue[(a^2+x^2)*D[y[x],{x,2}]+x*D[y[x],x]-n^2*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {n^4 x^5}{120 a^4}-\frac {n^2 x^5}{12 a^4}+\frac {3 x^5}{40 a^4}+\frac {n^2 x^3}{6 a^2}-\frac {x^3}{6 a^2}+x\right )+c_1 \left (\frac {n^4 x^4}{24 a^4}-\frac {n^2 x^4}{6 a^4}+\frac {n^2 x^2}{2 a^2}+1\right ) \]