Internal problem ID [6234]
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: 7.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
Solve \begin {gather*} \boxed {x \left (1-x \right ) y^{\prime \prime }+2 \left (1-x \right ) y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 54
Order:=8; dsolve(x*(1-x)*diff(y(x),x$2)+2*(1-x)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\left (-2\right ) x +2 x^{2}+\mathrm {O}\left (x^{8}\right )\right ) \ln \relax (x ) c_{2}+c_{1} \left (1-x +\mathrm {O}\left (x^{8}\right )\right ) x +\left (1-4 x^{2}+x^{3}+\frac {1}{3} x^{4}+\frac {1}{6} x^{5}+\frac {1}{10} x^{6}+\frac {1}{15} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}}{x} \]
✓ Solution by Mathematica
Time used: 0.341 (sec). Leaf size: 60
AsymptoticDSolveValue[x*(1-x)*y''[x]+2*(1-x)*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {3 x^6+5 x^5+10 x^4+30 x^3-150 x^2+30 x+30}{30 x}+2 (x-1) \log (x)\right )+c_2 (1-x) \]