Internal problem ID [13660]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional
Exercises. page 715
Problem number: 35.3 (b).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {x^{3} y^{\prime \prime }+y^{\prime } x^{2}+y=0} \] With the expansion point for the power series method at \(x = 2\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 54
Order:=6; dsolve(x^3*diff(y(x),x$2)+x^2*diff(y(x),x)+y(x)=0,y(x),type='series',x=2);
\[ y \left (x \right ) = \left (1-\frac {\left (x -2\right )^{2}}{16}+\frac {\left (x -2\right )^{3}}{24}-\frac {35 \left (x -2\right )^{4}}{1536}+\frac {89 \left (x -2\right )^{5}}{7680}\right ) y \left (2\right )+\left (x -2-\frac {\left (x -2\right )^{2}}{4}+\frac {\left (x -2\right )^{3}}{16}-\frac {\left (x -2\right )^{4}}{96}-\frac {19 \left (x -2\right )^{5}}{7680}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 87
AsymptoticDSolveValue[x^3*y''[x]+x^2*y'[x]+y[x]==0,y[x],{x,2,5}]
\[ y(x)\to c_1 \left (\frac {89 (x-2)^5}{7680}-\frac {35 (x-2)^4}{1536}+\frac {1}{24} (x-2)^3-\frac {1}{16} (x-2)^2+1\right )+c_2 \left (-\frac {19 (x-2)^5}{7680}-\frac {1}{96} (x-2)^4+\frac {1}{16} (x-2)^3-\frac {1}{4} (x-2)^2+x-2\right ) \]