Internal problem ID [6173]
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: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {2 x y^{\prime \prime }+\left (-2 x^{2}+1\right ) y^{\prime }-4 x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 36
Order:=8; dsolve(2*x*diff(y(x),x$2)+(1-2*x^2)*diff(y(x),x)-4*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2}{3} x^{2}+\frac {4}{21} x^{4}+\frac {8}{231} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 61
AsymptoticDSolveValue[2*x*y''[x]+(1-2*x^2)*y'[x]-4*x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^6}{48}+\frac {x^4}{8}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {8 x^6}{231}+\frac {4 x^4}{21}+\frac {2 x^2}{3}+1\right ) \]