68.1.9 problem Problem 1.7

Internal problem ID [14067]
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 1.7
Date solved : Monday, March 31, 2025 at 12:02:31 PM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\left (1-5 x \right ) y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 44

Order:=6; 
ode:=x*(1-x)*diff(diff(y(x),x),x)+(1-5*x)*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+4 x +9 x^{2}+16 x^{3}+25 x^{4}+36 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-4\right ) x -12 x^{2}-24 x^{3}-40 x^{4}-60 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 87

ode=x*(1-x)*D[y[x],{x,2}]+(1-5*x)*D[y[x],x]-4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (36 x^5+25 x^4+16 x^3+9 x^2+4 x+1\right )+c_2 \left (-60 x^5-40 x^4-24 x^3-12 x^2+\left (36 x^5+25 x^4+16 x^3+9 x^2+4 x+1\right ) \log (x)-4 x\right ) \]

✓ Sympy. Time used: 1.048 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x)*Derivative(y(x), (x, 2)) + (1 - 5*x)*Derivative(y(x), x) - 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {16 x^{5}}{225} + \frac {4 x^{4}}{9} - \frac {16 x^{3}}{9} + 4 x^{2} - 4 x + 1\right ) + O\left (x^{6}\right ) \]