Internal problem ID [1336]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS
I. Exercises 7.5. Page 358
Problem number: 47.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {6 x^{2} y^{\prime \prime }+x \left (6 x^{2}+1\right ) y^{\prime }+\left (9 x^{2}+1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 35
Order:=6; dsolve(6*x^2*diff(y(x),x$2)+x*(1+6*x^2)*diff(y(x),x)+(1+9*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-\frac {6}{13} x^{2}+\frac {36}{325} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 52
AsymptoticDSolveValue[6*x^2*y''[x]+x*(1+6*x^2)*y'[x]+(1+9*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {36 x^4}{325}-\frac {6 x^2}{13}+1\right )+c_2 \sqrt [3]{x} \left (\frac {x^4}{8}-\frac {x^2}{2}+1\right ) \]