Internal problem ID [6259]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number: 10.
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 (1+2 x \right ) y^{\prime }+\left (1+4 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.02 (sec). Leaf size: 41
Order:=8; dsolve(2*x^2*diff(y(x),x$2)-x*(1+2*x)*diff(y(x),x)+(1+4*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1-3 x +\frac {1}{2} x^{2}+\frac {1}{30} x^{3}+\frac {1}{280} x^{4}+\frac {1}{2520} x^{5}+\frac {1}{23760} x^{6}+\frac {1}{240240} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x \left (1-\frac {2}{3} x +\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 70
AsymptoticDSolveValue[2*x^2*y''[x]-x*(1+2*x)*y'[x]+(1+4*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \sqrt {x} \left (\frac {x^7}{240240}+\frac {x^6}{23760}+\frac {x^5}{2520}+\frac {x^4}{280}+\frac {x^3}{30}+\frac {x^2}{2}-3 x+1\right )+c_1 \left (1-\frac {2 x}{3}\right ) x \]