Internal problem ID [1369]
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: 17.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} \left (2+x \right ) y^{\prime \prime }+y^{\prime } x^{2}+\left (1-x \right ) 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: 69
Order:=6; dsolve(2*x^2*(2+x)*diff(y(x),x$2)+x^2*diff(y(x),x)+(1-x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \sqrt {x}\, \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+\frac {1}{4} x -\frac {1}{32} x^{2}+\frac {1}{128} x^{3}-\frac {5}{2048} x^{4}+\frac {7}{8192} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (-\frac {3}{4} x +\frac {3}{64} x^{2}-\frac {7}{768} x^{3}+\frac {61}{24576} x^{4}-\frac {391}{491520} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 134
AsymptoticDSolveValue[2*x^2*(2+x)*y''[x]+x^2*y'[x]+(1-x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {7 x^5}{8192}-\frac {5 x^4}{2048}+\frac {x^3}{128}-\frac {x^2}{32}+\frac {x}{4}+1\right )+c_2 \left (\sqrt {x} \left (-\frac {391 x^5}{491520}+\frac {61 x^4}{24576}-\frac {7 x^3}{768}+\frac {3 x^2}{64}-\frac {3 x}{4}\right )+\sqrt {x} \left (\frac {7 x^5}{8192}-\frac {5 x^4}{2048}+\frac {x^3}{128}-\frac {x^2}{32}+\frac {x}{4}+1\right ) \log (x)\right ) \]