Internal problem ID [13999]
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 (m).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (-x^{4}+x \right ) y^{\prime }+3 y x^{3}=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 35
Order:=6; dsolve(x^2*diff(y(x),x$2)+(x-x^4)*diff(y(x),x)+3*x^3*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-\frac {1}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 39
AsymptoticDSolveValue[x^2*y''[x]+(x-x^4)*y'[x]+3*x^3*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (1-\frac {x^3}{3}\right )+c_2 \left (\frac {x^3}{3}+\left (1-\frac {x^3}{3}\right ) \log (x)\right ) \]