Internal problem ID [1308]
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: 17.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {3 x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 45
Order:=6; dsolve(3*x^2*diff(y(x),x$2)+x*(1+x)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {1}{7} x +\frac {1}{70} x^{2}-\frac {1}{910} x^{3}+\frac {1}{14560} x^{4}-\frac {1}{276640} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{3} x +\frac {1}{18} x^{2}-\frac {1}{162} x^{3}+\frac {1}{1944} x^{4}-\frac {1}{29160} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 86
AsymptoticDSolveValue[3*x^2*y''[x]+x*(1+x)*y'[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (-\frac {x^5}{276640}+\frac {x^4}{14560}-\frac {x^3}{910}+\frac {x^2}{70}-\frac {x}{7}+1\right )+\frac {c_2 \left (-\frac {x^5}{29160}+\frac {x^4}{1944}-\frac {x^3}{162}+\frac {x^2}{18}-\frac {x}{3}+1\right )}{\sqrt [3]{x}} \]