Internal problem ID [6269]
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: 21.
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 y^{\prime } x^{2}-\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: 0.078 (sec). Leaf size: 55
Order:=8; dsolve(4*x^2*diff(y(x),x$2)+2*x^2*diff(y(x),x)-(x+3)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{2} \left (1-\frac {1}{6} x +\frac {1}{48} x^{2}-\frac {1}{480} x^{3}+\frac {1}{5760} x^{4}-\frac {1}{80640} x^{5}+\frac {1}{1290240} x^{6}-\frac {1}{23224320} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (-2+x -\frac {1}{4} x^{2}+\frac {1}{24} x^{3}-\frac {1}{192} x^{4}+\frac {1}{1920} x^{5}-\frac {1}{23040} x^{6}+\frac {1}{322560} x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.16 (sec). Leaf size: 130
AsymptoticDSolveValue[4*x^2*y''[x]+2*x^2*y'[x]-(x+3)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {x^{11/2}}{46080}-\frac {x^{9/2}}{3840}+\frac {x^{7/2}}{384}-\frac {x^{5/2}}{48}+\frac {x^{3/2}}{8}-\frac {\sqrt {x}}{2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {x^{15/2}}{1290240}-\frac {x^{13/2}}{80640}+\frac {x^{11/2}}{5760}-\frac {x^{9/2}}{480}+\frac {x^{7/2}}{48}-\frac {x^{5/2}}{6}+x^{3/2}\right ) \]