Internal problem ID [1440]
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
III. Exercises 7.7. Page 389
Problem number: 24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-x \left (-x^{2}+7\right ) y^{\prime }+12 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: 49
Order:=6; dsolve(x^2*diff(y(x),x$2)-x*(7-x^2)*diff(y(x),x)+12*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x^{2} \left (\left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{1} x^{4}+c_{2} \left (\ln \relax (x ) \left (72 x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (-144-72 x^{2}+54 x^{4}+\mathrm {O}\left (x^{6}\right )\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 55
AsymptoticDSolveValue[x^2*y''[x]-x*(7-x^2)*y'[x]+12*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^{10}}{8}-\frac {x^8}{2}+x^6\right )+c_1 \left (-\frac {1}{2} x^6 \log (x)-\frac {1}{4} \left (x^4-2 x^2-4\right ) x^2\right ) \]