Internal problem ID [13652]
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: 34.9 b(iv).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\sqrt {x}\, y^{\prime \prime }+y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 44
Order:=5; dsolve(sqrt(x)*diff(y(x),x$2)+diff(y(x),x)+x*y(x)=0,y(x),type='series',x=1);
\[ y \left (x \right ) = \left (1-\frac {\left (-1+x \right )^{2}}{2}+\frac {\left (-1+x \right )^{3}}{12}-\frac {\left (-1+x \right )^{4}}{96}\right ) y \left (1\right )+\left (-1+x -\frac {\left (-1+x \right )^{2}}{2}+\frac {\left (-1+x \right )^{3}}{12}-\frac {3 \left (-1+x \right )^{4}}{32}\right ) D\left (y \right )\left (1\right )+O\left (x^{5}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 69
AsymptoticDSolveValue[Sqrt[x]*y''[x]+y'[x]+x*y[x]==0,y[x],{x,1,4}]
\[ y(x)\to c_1 \left (-\frac {1}{96} (x-1)^4+\frac {1}{12} (x-1)^3-\frac {1}{2} (x-1)^2+1\right )+c_2 \left (-\frac {3}{32} (x-1)^4+\frac {1}{12} (x-1)^3-\frac {1}{2} (x-1)^2+x-1\right ) \]