Internal problem ID [1393]
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: 41.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-9 x^{2}+5\right ) y^{\prime }+\left (-3 x^{2}+4\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 51
Order:=6; dsolve(x^2*(1-2*x^2)*diff(y(x),x$2)+x*(5-9*x^2)*diff(y(x),x)+(4-3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {3}{4} x^{2}-\frac {9}{64} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x^{2}-\frac {21}{128} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 71
AsymptoticDSolveValue[x^2*(1-2*x^2)*y''[x]+x*(5-9*x^2)*y'[x]+(4-3*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_1 \left (-\frac {9 x^4}{64}-\frac {3 x^2}{4}+1\right )}{x^2}+c_2 \left (\frac {\frac {x^2}{2}-\frac {21 x^4}{128}}{x^2}+\frac {\left (-\frac {9 x^4}{64}-\frac {3 x^2}{4}+1\right ) \log (x)}{x^2}\right ) \]