Internal problem ID [13674]
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: 35.4 (h).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {2 x^{2} y^{\prime \prime }+\left (-2 x^{3}+5 x \right ) y^{\prime }+\left (-x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 31
Order:=6; dsolve(2*x^2*diff(y(x),x$2)+(5*x-2*x^3)*diff(y(x),x)+(1-x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{6} x^{2}-\frac {1}{56} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \sqrt {x}+c_{2} \left (1+\operatorname {O}\left (x^{6}\right )\right ) x}{x^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 34
AsymptoticDSolveValue[2*x^2*y''[x]+(5*x-2*x^3)*y'[x]+(1-x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_2 \left (-\frac {x^4}{56}-\frac {x^2}{6}+1\right )}{x}+\frac {c_1}{\sqrt {x}} \]