Internal problem ID [4848]
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. 6.3.1 page
250
Problem number: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+2 y^{\prime }+4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.026 (sec). Leaf size: 64
Order:=6; dsolve(x*diff(y(x),x$2)+2*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} \left (1-2 x +\frac {4}{3} x^{2}-\frac {4}{9} x^{3}+\frac {4}{45} x^{4}-\frac {8}{675} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) x +c_{2} \left (\ln \relax (x ) \left (\left (-4\right ) x +8 x^{2}-\frac {16}{3} x^{3}+\frac {16}{9} x^{4}-\frac {16}{45} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (1-12 x^{2}+\frac {112}{9} x^{3}-\frac {140}{27} x^{4}+\frac {808}{675} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 85
AsymptoticDSolveValue[x*y''[x]+2*y'[x]+4*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {4 x^4}{45}-\frac {4 x^3}{9}+\frac {4 x^2}{3}-2 x+1\right )+c_1 \left (\frac {4}{9} \left (4 x^3-12 x^2+18 x-9\right ) \log (x)-\frac {188 x^4-480 x^3+540 x^2-108 x-27}{27 x}\right ) \]