Internal problem ID [6179]
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: 16.
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 }+x \left (4 x -1\right ) y^{\prime }+2 \left (3 x -1\right ) 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: 55
Order:=8; dsolve(2*x^2*diff(y(x),x$2)+x*(4*x-1)*diff(y(x),x)+2*(3*x-1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {5}{2}} \left (1-2 x +2 x^{2}-\frac {4}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{15} x^{5}+\frac {4}{45} x^{6}-\frac {8}{315} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{1} \left (1+\frac {4}{3} x +\frac {16}{3} x^{2}-\frac {64}{3} x^{3}+\frac {256}{9} x^{4}-\frac {1024}{45} x^{5}+\frac {4096}{315} x^{6}-\frac {16384}{2835} x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 112
AsymptoticDSolveValue[2*x^2*y''[x]+x*(4*x-1)*y'[x]+2*(3*x-1)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {8 x^7}{315}+\frac {4 x^6}{45}-\frac {4 x^5}{15}+\frac {2 x^4}{3}-\frac {4 x^3}{3}+2 x^2-2 x+1\right ) x^2+\frac {c_2 \left (-\frac {16384 x^7}{2835}+\frac {4096 x^6}{315}-\frac {1024 x^5}{45}+\frac {256 x^4}{9}-\frac {64 x^3}{3}+\frac {16 x^2}{3}+\frac {4 x}{3}+1\right )}{\sqrt {x}} \]