Internal problem ID [5712]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Section 4.5. More on Regular
Singular Points. Page 183
Problem number: 3(c).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-y^{\prime }+4 y x^{3}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 28
Order:=8; dsolve(x*diff(y(x),x$2)-diff(y(x),x)+4*x^3*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{2} \left (1-\frac {1}{6} x^{4}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (-2+x^{4}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 30
AsymptoticDSolveValue[x*y''[x]-y'[x]+4*x^3*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (1-\frac {x^4}{2}\right )+c_2 \left (x^2-\frac {x^6}{6}\right ) \]