Internal problem ID [4218]
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: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.025 (sec). Leaf size: 42
Order:=6; dsolve(2*x*(1-x)*diff(y(x),x$2)+x*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \ln \relax (x ) \left (\frac {1}{2} x +\mathrm {O}\left (x^{6}\right )\right ) c_{2}+c_{1} x \left (1+\mathrm {O}\left (x^{6}\right )\right )+\left (1-\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{32} x^{3}+\frac {5}{384} x^{4}+\frac {7}{1024} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.046 (sec). Leaf size: 43
AsymptoticDSolveValue[2*x*(1-x)*y''[x]+x*y'[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{384} \left (5 x^4+12 x^3+48 x^2-768 x+384\right )+\frac {1}{2} x \log (x)\right )+c_2 x \]