Internal problem ID [4213]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter VII, Solutions in series. Examples XV. page 194
Problem number: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
Solve \begin {gather*} \boxed {4 x \left (1-x \right ) y^{\prime \prime }-4 y^{\prime }-y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.027 (sec). Leaf size: 60
Order:=6; dsolve(4*x*(1-x)*diff(y(x),x$2)-4*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{2} \left (1+\frac {3}{4} x +\frac {75}{128} x^{2}+\frac {245}{512} x^{3}+\frac {6615}{16384} x^{4}+\frac {22869}{65536} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (\frac {1}{16} x^{2}+\frac {3}{64} x^{3}+\frac {75}{2048} x^{4}+\frac {245}{8192} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (-2+\frac {1}{2} x +\frac {1}{2} x^{2}+\frac {3}{8} x^{3}+\frac {2415}{8192} x^{4}+\frac {23779}{98304} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.045 (sec). Leaf size: 86
AsymptoticDSolveValue[4*x*(1-x)*y''[x]-4*y'[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {135 x^4+192 x^3+256 x^2-4096 x+16384}{16384}-\frac {x^2 \left (75 x^2+96 x+128\right ) \log (x)}{4096}\right )+c_2 \left (\frac {6615 x^6}{16384}+\frac {245 x^5}{512}+\frac {75 x^4}{128}+\frac {3 x^3}{4}+x^2\right ) \]