Internal problem ID [6201]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises
page 373
Problem number: 7.
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 (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: 0.063 (sec). Leaf size: 47
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*(x-1)*diff(y(x),x)+(1-x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+\mathrm {O}\left (x^{8}\right )\right )+\left (-x +\frac {1}{4} x^{2}-\frac {1}{18} x^{3}+\frac {1}{96} x^{4}-\frac {1}{600} x^{5}+\frac {1}{4320} x^{6}-\frac {1}{35280} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 64
AsymptoticDSolveValue[x^2*y''[x]+x*(x-1)*y'[x]+(1-x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (x \left (-\frac {x^7}{35280}+\frac {x^6}{4320}-\frac {x^5}{600}+\frac {x^4}{96}-\frac {x^3}{18}+\frac {x^2}{4}-x\right )+x \log (x)\right )+c_1 x \]