Internal problem ID [1422]
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: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x \left (2+x \right ) y^{\prime }-\left (-3 x +2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 63
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*(2+x)*diff(y(x),x)-(2-3*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{3} \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (\left (-6\right ) x^{3}+6 x^{4}-3 x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (12+6 x +6 x^{2}-11 x^{3}+5 x^{4}-x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 74
AsymptoticDSolveValue[x^2*y''[x]+x*(2+x)*y'[x]-(2-3*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x^4-3 x^3+2 x^2+2 x+4}{4 x^2}+\frac {1}{2} (x-1) x \log (x)\right )+c_2 \left (\frac {x^5}{24}-\frac {x^4}{6}+\frac {x^3}{2}-x^2+x\right ) \]