Internal problem ID [1313]
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: 22.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (x +4\right ) y^{\prime \prime }-x \left (-3 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: 0.013 (sec). Leaf size: 45
Order:=6; dsolve(x^2*(4+x)*diff(y(x),x$2)-x*(1-3*x)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {9}{16} x +\frac {117}{512} x^{2}-\frac {663}{8192} x^{3}+\frac {13923}{524288} x^{4}-\frac {69615}{8388608} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x \left (1-\frac {3}{7} x +\frac {12}{77} x^{2}-\frac {4}{77} x^{3}+\frac {24}{1463} x^{4}-\frac {24}{4807} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 86
AsymptoticDSolveValue[x^2*(4+x)*y''[x]-x*(1-3*x)*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (-\frac {24 x^5}{4807}+\frac {24 x^4}{1463}-\frac {4 x^3}{77}+\frac {12 x^2}{77}-\frac {3 x}{7}+1\right )+c_2 \sqrt [4]{x} \left (-\frac {69615 x^5}{8388608}+\frac {13923 x^4}{524288}-\frac {663 x^3}{8192}+\frac {117 x^2}{512}-\frac {9 x}{16}+1\right ) \]