Internal problem ID [1411]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS
II. Exercises 7.6. Page 374
Problem number: 64.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (1-x \right )^{2} y^{\prime \prime }-x \left (-3 x^{2}+2 x +1\right ) y^{\prime }+y \left (x^{2}+1\right )=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: 45
Order:=6; dsolve(x^2*(1-x)^2*diff(y(x),x$2)-x*(1+2*x-3*x^2)*diff(y(x),x)+(1+x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (6 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1\right ) x \left (c_{2} \ln \relax (x )+c_{1}\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 64
AsymptoticDSolveValue[x^2*(1-x)^2*y''[x]-x*(1+2*x-3*x^2)*y'[x]+(1+x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (6 x^5+5 x^4+4 x^3+3 x^2+2 x+1\right )+c_2 x \left (6 x^5+5 x^4+4 x^3+3 x^2+2 x+1\right ) \log (x) \]