Internal problem ID [1303]
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: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (x^{2}+x +3\right ) y^{\prime \prime }+\left (-x^{2}+x +4\right ) y^{\prime }+y x=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 42
Order:=6; dsolve(x*(3+x+x^2)*diff(y(x),x$2)+(4+x-x^2)*diff(y(x),x)+x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} \left (1-\frac {1}{14} x^{2}+\frac {1}{105} x^{3}-\frac {1}{3640} x^{4}-\frac {23}{54600} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) x^{\frac {1}{3}}+c_{1} \left (1-\frac {1}{18} x -\frac {71}{405} x^{2}+\frac {719}{34992} x^{3}-\frac {1678}{1082565} x^{4}-\frac {513547}{992023200} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 80
AsymptoticDSolveValue[x*(3+x+x^2)*y''[x]+(4+x-x^2)*y'[x]+x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {23 x^5}{54600}-\frac {x^4}{3640}+\frac {x^3}{105}-\frac {x^2}{14}+1\right )+\frac {c_2 \left (-\frac {513547 x^5}{992023200}-\frac {1678 x^4}{1082565}+\frac {719 x^3}{34992}-\frac {71 x^2}{405}-\frac {x}{18}+1\right )}{\sqrt [3]{x}} \]