Internal problem ID [4217]
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: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
Solve \begin {gather*} \boxed {x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.02 (sec). Leaf size: 34
Order:=6; dsolve(x*(1-x)*diff(y(x),x$2)+(3/2-2*x)*diff(y(x),x)-1/4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} \left (1+\frac {1}{6} x +\frac {3}{40} x^{2}+\frac {5}{112} x^{3}+\frac {35}{1152} x^{4}+\frac {63}{2816} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \sqrt {x}+c_{1} \left (1+\mathrm {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 50
AsymptoticDSolveValue[x*(1-x)*y''[x]+(3/2-2*x)*y'[x]-1/4*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {63 x^5}{2816}+\frac {35 x^4}{1152}+\frac {5 x^3}{112}+\frac {3 x^2}{40}+\frac {x}{6}+1\right )+\frac {c_2}{\sqrt {x}} \]