Internal problem ID [6230]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a
Positive Integer: Logarithmic Case. Exercises page 384
Problem number: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {2 x y^{\prime \prime }+6 y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 74
Order:=8; dsolve(2*x*diff(y(x),x$2)+6*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} \left (1-\frac {1}{6} x +\frac {1}{96} x^{2}-\frac {1}{2880} x^{3}+\frac {1}{138240} x^{4}-\frac {1}{9676800} x^{5}+\frac {1}{928972800} x^{6}-\frac {1}{117050572800} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) x^{2}+c_{2} \left (\ln \relax (x ) \left (\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{384} x^{4}-\frac {1}{11520} x^{5}+\frac {1}{552960} x^{6}-\frac {1}{38707200} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (-2-x +\frac {1}{18} x^{3}-\frac {25}{4608} x^{4}+\frac {157}{691200} x^{5}-\frac {91}{16588800} x^{6}+\frac {709}{8128512000} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 114
AsymptoticDSolveValue[2*x*y''[x]+6*y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {x^6}{928972800}-\frac {x^5}{9676800}+\frac {x^4}{138240}-\frac {x^3}{2880}+\frac {x^2}{96}-\frac {x}{6}+1\right )+c_1 \left (\frac {53 x^6-2244 x^5+55800 x^4-633600 x^3+1036800 x^2+8294400 x+16588800}{16588800 x^2}-\frac {\left (x^4-48 x^3+1440 x^2-23040 x+138240\right ) \log (x)}{1105920}\right ) \]