Internal problem ID [4819]
Book: A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G.
Zill. 9th edition. Brooks/Cole. CA, USA.
Section: Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.2 page
239
Problem number: 17.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {4 x y^{\prime \prime }+\frac {y^{\prime }}{2}+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 44
Order:=6; dsolve(4*x*diff(y(x),x$2)+1/2*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {7}{8}} \left (1-\frac {2}{15} x +\frac {2}{345} x^{2}-\frac {4}{32085} x^{3}+\frac {2}{1251315} x^{4}-\frac {4}{294059025} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x +\frac {2}{9} x^{2}-\frac {4}{459} x^{3}+\frac {2}{11475} x^{4}-\frac {4}{1893375} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 83
AsymptoticDSolveValue[4*x*y''[x]+1/2*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (-\frac {4 x^5}{1893375}+\frac {2 x^4}{11475}-\frac {4 x^3}{459}+\frac {2 x^2}{9}-2 x+1\right )+c_1 x^{7/8} \left (-\frac {4 x^5}{294059025}+\frac {2 x^4}{1251315}-\frac {4 x^3}{32085}+\frac {2 x^2}{345}-\frac {2 x}{15}+1\right ) \]