Internal problem ID [4862]
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. Chapter 6 review exercises.
page 253
Problem number: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {2 x y^{\prime \prime }+y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 44
Order:=6; dsolve(2*x*diff(y(x),x$2)+diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{3} x +\frac {1}{30} x^{2}-\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}-\frac {1}{1247400} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (1-x +\frac {1}{6} x^{2}-\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}-\frac {1}{113400} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 83
AsymptoticDSolveValue[2*x*y''[x]+y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^5}{1247400}+\frac {x^4}{22680}-\frac {x^3}{630}+\frac {x^2}{30}-\frac {x}{3}+1\right )+c_2 \left (-\frac {x^5}{113400}+\frac {x^4}{2520}-\frac {x^3}{90}+\frac {x^2}{6}-x+1\right ) \]