Internal problem ID [1420]
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
III. Exercises 7.7. Page 389
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (x +1\right ) y^{\prime \prime }+y^{\prime } x +y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 60
Order:=6; dsolve(x*(1+x)*diff(y(x),x$2)+x*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \ln \relax (x ) \left (-x +x^{2}-\frac {5}{6} x^{3}+\frac {25}{36} x^{4}-\frac {85}{144} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}+c_{1} x \left (1-x +\frac {5}{6} x^{2}-\frac {25}{36} x^{3}+\frac {85}{144} x^{4}-\frac {221}{432} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (1-x +\frac {1}{2} x^{2}-\frac {7}{18} x^{3}+\frac {145}{432} x^{4}-\frac {257}{864} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.028 (sec). Leaf size: 85
AsymptoticDSolveValue[x*(1+x)*y''[x]+x*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{36} x \left (25 x^3-30 x^2+36 x-36\right ) \log (x)+\frac {1}{432} \left (-455 x^4+552 x^3-648 x^2+432 x+432\right )\right )+c_2 \left (\frac {85 x^5}{144}-\frac {25 x^4}{36}+\frac {5 x^3}{6}-x^2+x\right ) \]