Internal problem ID [12088]
Book: Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham,
S.R.Otto. Cambridge Univ. Press 2003
Section: Chapter 1 VARIABLE COEFFICIENT, SECOND ORDER DIFFERENTIAL
EQUATIONS. Problems page 28
Problem number: Problem 1.12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {2 y^{\prime \prime } x^{2}+x \left (1-x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 45
Order:=6; dsolve(2*x^2*diff(y(x),x$2)+x*(1-x)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \frac {c_{2} x^{\frac {3}{2}} \left (1+\frac {1}{5} x +\frac {1}{35} x^{2}+\frac {1}{315} x^{3}+\frac {1}{3465} x^{4}+\frac {1}{45045} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 86
AsymptoticDSolveValue[2*x^2*y''[x]+x*(1-x)*y'[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (\frac {x^5}{45045}+\frac {x^4}{3465}+\frac {x^3}{315}+\frac {x^2}{35}+\frac {x}{5}+1\right )+\frac {c_2 \left (\frac {x^5}{3840}+\frac {x^4}{384}+\frac {x^3}{48}+\frac {x^2}{8}+\frac {x}{2}+1\right )}{\sqrt {x}} \]