Internal problem ID [4771]
Book: Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section: Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page
300
Problem number: 7.3.8 (e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x^{2} y^{\prime }+y x^{2}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 44
Order:=6; dsolve(x^2*diff(y(x),x$2)+x^2*diff(y(x),x)+x^2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{120} x^{5}\right ) y \relax (0)+\left (x -\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 56
AsymptoticDSolveValue[x^2*y''[x]+x^2*y'[x]+x^2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^2}{2}+x\right )+c_1 \left (-\frac {x^5}{120}+\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]