Internal problem ID [1378]
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: 26.
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 +1\right ) y^{\prime \prime }+x \left (1+2 x \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: 0.017 (sec). Leaf size: 71
Order:=8; dsolve(x^2*(1+x)*diff(y(x),x$2)+x*(1+2*x)*diff(y(x),x)+x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-x +\frac {3}{4} x^{2}-\frac {7}{12} x^{3}+\frac {91}{192} x^{4}-\frac {637}{1600} x^{5}+\frac {19747}{57600} x^{6}-\frac {17329}{57600} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (x -\frac {3}{4} x^{2}+\frac {5}{9} x^{3}-\frac {499}{1152} x^{4}+\frac {16919}{48000} x^{5}-\frac {56861}{192000} x^{6}+\frac {1027717}{4032000} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 151
AsymptoticDSolveValue[x^2*(1+x)*y''[x]+x*(1+2*x)*y'[x]+x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {17329 x^7}{57600}+\frac {19747 x^6}{57600}-\frac {637 x^5}{1600}+\frac {91 x^4}{192}-\frac {7 x^3}{12}+\frac {3 x^2}{4}-x+1\right )+c_2 \left (\frac {1027717 x^7}{4032000}-\frac {56861 x^6}{192000}+\frac {16919 x^5}{48000}-\frac {499 x^4}{1152}+\frac {5 x^3}{9}-\frac {3 x^2}{4}+\left (-\frac {17329 x^7}{57600}+\frac {19747 x^6}{57600}-\frac {637 x^5}{1600}+\frac {91 x^4}{192}-\frac {7 x^3}{12}+\frac {3 x^2}{4}-x+1\right ) \log (x)+x\right ) \]