Internal problem ID [6178]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots
Nonintegral. Exercises page 365
Problem number: 15.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} y^{\prime \prime }-3 x \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: 0.062 (sec). Leaf size: 55
Order:=8; dsolve(2*x^2*diff(y(x),x$2)-3*x*(1-x)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1+\frac {3}{2} x -\frac {27}{8} x^{2}+\frac {45}{16} x^{3}-\frac {189}{128} x^{4}+\frac {729}{1280} x^{5}-\frac {891}{5120} x^{6}+\frac {3159}{71680} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{2} \left (1-\frac {6}{5} x +\frac {27}{35} x^{2}-\frac {12}{35} x^{3}+\frac {9}{77} x^{4}-\frac {162}{5005} x^{5}+\frac {27}{3575} x^{6}-\frac {648}{425425} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 116
AsymptoticDSolveValue[2*x^2*y''[x]-3*x*(1-x)*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {648 x^7}{425425}+\frac {27 x^6}{3575}-\frac {162 x^5}{5005}+\frac {9 x^4}{77}-\frac {12 x^3}{35}+\frac {27 x^2}{35}-\frac {6 x}{5}+1\right ) x^2+c_2 \left (\frac {3159 x^7}{71680}-\frac {891 x^6}{5120}+\frac {729 x^5}{1280}-\frac {189 x^4}{128}+\frac {45 x^3}{16}-\frac {27 x^2}{8}+\frac {3 x}{2}+1\right ) \sqrt {x} \]