Internal problem ID [6241]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a
Positive Integer: Logarithmic Case. Exercises page 384
Problem number: 14.
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 +1\right ) y^{\prime }-\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: 0.094 (sec). Leaf size: 75
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*(1-2*x)*diff(y(x),x)-(x+1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{2} \left (1+x +\frac {5}{8} x^{2}+\frac {7}{24} x^{3}+\frac {7}{64} x^{4}+\frac {11}{320} x^{5}+\frac {143}{15360} x^{6}+\frac {143}{64512} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \relax (x ) \left (-x^{2}-x^{3}-\frac {5}{8} x^{4}-\frac {7}{24} x^{5}-\frac {7}{64} x^{6}-\frac {11}{320} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (-2-2 x +\frac {2}{3} x^{3}+\frac {61}{96} x^{4}+\frac {59}{160} x^{5}+\frac {919}{5760} x^{6}+\frac {449}{8064} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.164 (sec). Leaf size: 115
AsymptoticDSolveValue[x^2*y''[x]+x*(1-2*x)*y'[x]-(x+1)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {1}{384} x \left (21 x^4+56 x^3+120 x^2+192 x+192\right ) \log (x)-\frac {617 x^6+1482 x^5+2730 x^4+3360 x^3+1440 x^2-5760 x-5760}{5760 x}\right )+c_2 \left (\frac {143 x^7}{15360}+\frac {11 x^6}{320}+\frac {7 x^5}{64}+\frac {7 x^4}{24}+\frac {5 x^3}{8}+x^2+x\right ) \]