Internal problem ID [4204]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XV. page 194
Problem number: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+y^{\prime }+p x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 59
Order:=6; dsolve(x*diff(y(x),x$2)+diff(y(x),x)+p*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {1}{4} p \,x^{2}+\frac {1}{64} p^{2} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {p}{4} x^{2}-\frac {3}{128} p^{2} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 72
AsymptoticDSolveValue[x*y''[x]+y'[x]+p*x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {p^2 x^4}{64}-\frac {p x^2}{4}+1\right )+c_2 \left (-\frac {3}{128} p^2 x^4+\left (\frac {p^2 x^4}{64}-\frac {p x^2}{4}+1\right ) \log (x)+\frac {p x^2}{4}\right ) \]