Internal problem ID [1355]
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: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {x^{2} \left (x^{2}+2 x +1\right ) y^{\prime \prime }+x \left (4 x^{2}+3 x +1\right ) y^{\prime }-x \left (1-2 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 71
Order:=8; dsolve(x^2*(1+2*x+x^2)*diff(y(x),x$2)+x*(1+3*x+4*x^2)*diff(y(x),x)-x*(1-2*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1+x -x^{2}+\frac {1}{3} x^{3}+\frac {1}{3} x^{4}-\frac {11}{15} x^{5}+\frac {37}{45} x^{6}-\frac {209}{315} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (\left (-3\right ) x +\frac {1}{2} x^{2}+\frac {31}{18} x^{3}-\frac {91}{36} x^{4}+\frac {1897}{900} x^{5}-\frac {301}{300} x^{6}-\frac {3901}{14700} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 145
AsymptoticDSolveValue[x^2*(1+2*x+x^2)*y''[x]+x*(1+3*x+4*x^2)*y'[x]-x*(1-2*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {209 x^7}{315}+\frac {37 x^6}{45}-\frac {11 x^5}{15}+\frac {x^4}{3}+\frac {x^3}{3}-x^2+x+1\right )+c_2 \left (-\frac {3901 x^7}{14700}-\frac {301 x^6}{300}+\frac {1897 x^5}{900}-\frac {91 x^4}{36}+\frac {31 x^3}{18}+\frac {x^2}{2}+\left (-\frac {209 x^7}{315}+\frac {37 x^6}{45}-\frac {11 x^5}{15}+\frac {x^4}{3}+\frac {x^3}{3}-x^2+x+1\right ) \log (x)-3 x\right ) \]