Internal problem ID [5494]
Book: Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag.
Springer October 29, 1999
Section: Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS.
page 136
Problem number: 3.25 v=1/2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}+\frac {1}{4}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2+(1/2)^2)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} x^{-\frac {i}{2}} \left (1+\left (-\frac {1}{5}-\frac {i}{10}\right ) x^{2}+\left (\frac {7}{680}+\frac {3 i}{340}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {i}{2}} \left (1+\left (-\frac {1}{5}+\frac {i}{10}\right ) x^{2}+\left (\frac {7}{680}-\frac {3 i}{340}\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 66
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x^2+1/4)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \left (\frac {7}{680}+\frac {3 i}{340}\right ) c_2 x^{-\frac {i}{2}} \left (x^4-(16-4 i) x^2+(56-48 i)\right )+\left (\frac {7}{680}-\frac {3 i}{340}\right ) c_1 x^{\frac {i}{2}} \left (x^4-(16+4 i) x^2+(56+48 i)\right ) \]