Internal problem ID [6220]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a
Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x \left (1-x \right ) y^{\prime \prime }-3 y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 38
Order:=8; dsolve(x*(1-x)*diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=1);
\[ y \relax (x ) = c_{1} \left (1+\frac {2}{3} \left (x -1\right )+\frac {1}{6} \left (x -1\right )^{2}+\mathrm {O}\left (\left (x -1\right )^{8}\right )\right )+\frac {c_{2} \left (-2-8 \left (x -1\right )-12 \left (x -1\right )^{2}-8 \left (x -1\right )^{3}-2 \left (x -1\right )^{4}+\mathrm {O}\left (\left (x -1\right )^{8}\right )\right )}{\left (x -1\right )^{2}} \]
✓ Solution by Mathematica
Time used: 0.358 (sec). Leaf size: 52
AsymptoticDSolveValue[x*(1-x)*y''[x]-3*y'[x]+2*y[x]==0,y[x],{x,1,7}]
\[ y(x)\to c_1 \left ((x-1)^2+4 (x-1)+\frac {4}{x-1}+\frac {1}{(x-1)^2}+6\right )+c_2 \left (\frac {1}{6} (x-1)^2+\frac {2 (x-1)}{3}+1\right ) \]