Internal problem ID [4221]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XVI. page 220
Problem number: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {2 x \left (1-x \right ) y^{\prime \prime }+y^{\prime }+4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 38
Order:=6; dsolve(2*x*(1-x)*diff(y(x),x$2)+diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1-\frac {3}{2} x +\frac {3}{8} x^{2}+\frac {1}{16} x^{3}+\frac {3}{128} x^{4}+\frac {3}{256} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (1-4 x +\frac {8}{3} x^{2}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 62
AsymptoticDSolveValue[2*x*(1-x)*y''[x]+y'[x]+4*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {8 x^2}{3}-4 x+1\right )+c_1 \sqrt {x} \left (\frac {3 x^5}{256}+\frac {3 x^4}{128}+\frac {x^3}{16}+\frac {3 x^2}{8}-\frac {3 x}{2}+1\right ) \]