Internal problem ID [1372]
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: 20.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x^{2} \left (-4 x +1\right ) y^{\prime \prime }+3 x \left (1-6 x \right ) y^{\prime }+\left (1-12 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 69
Order:=6; dsolve(x^2*(1-4*x)*diff(y(x),x$2)+3*x*(1-6*x)*diff(y(x),x)+(1-12*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+2 x +6 x^{2}+20 x^{3}+70 x^{4}+252 x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (2 x +7 x^{2}+\frac {74}{3} x^{3}+\frac {533}{6} x^{4}+\frac {1627}{5} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 104
AsymptoticDSolveValue[x^2*(1-4*x)*y''[x]+3*x*(1-6*x)*y'[x]+(1-12*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_1 \left (252 x^5+70 x^4+20 x^3+6 x^2+2 x+1\right )}{x}+c_2 \left (\frac {\frac {1627 x^5}{5}+\frac {533 x^4}{6}+\frac {74 x^3}{3}+7 x^2+2 x}{x}+\frac {\left (252 x^5+70 x^4+20 x^3+6 x^2+2 x+1\right ) \log (x)}{x}\right ) \]