Internal problem ID [5290]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number: 1(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.023 (sec). Leaf size: 70
Order:=8; dsolve(x*diff(y(x),x$2)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-2 x +\frac {4}{3} x^{2}-\frac {4}{9} x^{3}+\frac {4}{45} x^{4}-\frac {8}{675} x^{5}+\frac {16}{14175} x^{6}-\frac {8}{99225} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \relax (x ) \left (\left (-4\right ) x +8 x^{2}-\frac {16}{3} x^{3}+\frac {16}{9} x^{4}-\frac {16}{45} x^{5}+\frac {32}{675} x^{6}-\frac {64}{14175} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (1-12 x^{2}+\frac {112}{9} x^{3}-\frac {140}{27} x^{4}+\frac {808}{675} x^{5}-\frac {1792}{10125} x^{6}+\frac {9056}{496125} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 119
AsymptoticDSolveValue[x*y''[x]+4*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {4}{675} x \left (8 x^5-60 x^4+300 x^3-900 x^2+1350 x-675\right ) \log (x)+\frac {-2272 x^6+15720 x^5-70500 x^4+180000 x^3-202500 x^2+40500 x+10125}{10125}\right )+c_2 \left (\frac {16 x^7}{14175}-\frac {8 x^6}{675}+\frac {4 x^5}{45}-\frac {4 x^4}{9}+\frac {4 x^3}{3}-2 x^2+x\right ) \]