Internal problem ID [6260]
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: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }-2 x \left (2+x \right ) y^{\prime }+\left (x +3\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.03 (sec). Leaf size: 55
Order:=8; dsolve(4*x^2*diff(y(x),x$2)-2*x*(2+x)*diff(y(x),x)+(3+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \sqrt {x}\, \left (x \left (1+\frac {1}{4} x +\frac {1}{24} x^{2}+\frac {1}{192} x^{3}+\frac {1}{1920} x^{4}+\frac {1}{23040} x^{5}+\frac {1}{322560} x^{6}+\frac {1}{5160960} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{1}+\left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\frac {1}{46080} x^{6}+\frac {1}{645120} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.083 (sec). Leaf size: 130
AsymptoticDSolveValue[4*x^2*y''[x]-2*x*(2+x)*y'[x]+(3+x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {x^{13/2}}{46080}+\frac {x^{11/2}}{3840}+\frac {x^{9/2}}{384}+\frac {x^{7/2}}{48}+\frac {x^{5/2}}{8}+\frac {x^{3/2}}{2}+\sqrt {x}\right )+c_2 \left (\frac {x^{15/2}}{322560}+\frac {x^{13/2}}{23040}+\frac {x^{11/2}}{1920}+\frac {x^{9/2}}{192}+\frac {x^{7/2}}{24}+\frac {x^{5/2}}{4}+x^{3/2}\right ) \]