Internal problem ID [1398]
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: 46.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (-2 x +3\right ) y^{\prime }+\left (1+2 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 57
Order:=6; dsolve(x^2*(1-x)*diff(y(x),x$2)+x*(3-2*x)*diff(y(x),x)+(1+2*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (3 x -3 x^{2}+\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}+\left (1-2 x +x^{2}+\mathrm {O}\left (x^{6}\right )\right ) \left (\ln \relax (x ) c_{2}+c_{1}\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 70
AsymptoticDSolveValue[x^2*(1-x)*y''[x]+x*(3-2*x)*y'[x]+(1+2*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_1 \left (x^2-2 x+1\right )}{x}+c_2 \left (\frac {\left (x^2-2 x+1\right ) \log (x)}{x}+\frac {\frac {x^5}{30}+\frac {x^4}{12}+\frac {x^3}{3}-3 x^2+3 x}{x}\right ) \]