Internal problem ID [1402]
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: 50.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {9 x^{2} y^{\prime \prime }+3 x \left (-x^{2}+1\right ) y^{\prime }+\left (7 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: 0.015 (sec). Leaf size: 47
Order:=6; dsolve(9*x^2*diff(y(x),x$2)+3*x*(1-x^2)*diff(y(x),x)+(1+7*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1-\frac {1}{6} x^{2}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {1}{288} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) x^{\frac {1}{3}} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 63
AsymptoticDSolveValue[9*x^2*y''[x]+3*x*(1-x^2)*y'[x]+(1+7*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [3]{x} \left (1-\frac {x^2}{6}\right )+c_2 \left (\sqrt [3]{x} \left (1-\frac {x^2}{6}\right ) \log (x)+\sqrt [3]{x} \left (\frac {x^2}{4}-\frac {x^4}{288}\right )\right ) \]