Internal problem ID [1387]
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: 35.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {3 x^{2} \left (x^{2}+3\right ) y^{\prime \prime }+x \left (11 x^{2}+3\right ) y^{\prime }+\left (5 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: 0.011 (sec). Leaf size: 51
Order:=6; dsolve(3*x^2*(3+x^2)*diff(y(x),x$2)+x*(3+11*x^2)*diff(y(x),x)+(1+5*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x^{\frac {1}{3}} \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {2}{9} x^{2}+\frac {5}{81} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (-\frac {1}{18} x^{2}+\frac {1}{54} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 42
AsymptoticDSolveValue[3*x^2*(3+x^2)*y'[x]+x*(3+11*x^2)*y'[x]+(1+5*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_1 \left (-\frac {1139 x^5}{405}+\frac {53 x^4}{81}+\frac {7 x^3}{9}-\frac {11 x^2}{9}+x+1\right )}{\sqrt [3]{x}} \]