Internal problem ID [13643]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises.
page 678
Problem number: 34.7 (e).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {\sqrt {x}\, y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 49
Order:=6; dsolve(sqrt(x)*diff(y(x),x$2)+y(x)=0,y(x),type='series',x=1);
\[ y \left (x \right ) = \left (1-\frac {\left (-1+x \right )^{2}}{2}+\frac {\left (-1+x \right )^{3}}{12}+\frac {\left (-1+x \right )^{4}}{96}-\frac {\left (-1+x \right )^{5}}{960}\right ) y \left (1\right )+\left (-1+x -\frac {\left (-1+x \right )^{3}}{6}+\frac {\left (-1+x \right )^{4}}{24}-\frac {\left (-1+x \right )^{5}}{96}\right ) D\left (y \right )\left (1\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 78
AsymptoticDSolveValue[Sqrt[x]*y''[x]+y[x]==0,y[x],{x,1,5}]
\[ y(x)\to c_1 \left (-\frac {1}{960} (x-1)^5+\frac {1}{96} (x-1)^4+\frac {1}{12} (x-1)^3-\frac {1}{2} (x-1)^2+1\right )+c_2 \left (-\frac {1}{96} (x-1)^5+\frac {1}{24} (x-1)^4-\frac {1}{6} (x-1)^3+x-1\right ) \]