68.17.23 problem 23

Internal problem ID [17841]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 23
Date solved : Thursday, October 02, 2025 at 02:28:47 PM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 36

Order:=6; 
ode:=x*(1-x)*diff(diff(y(x),x),x)+(1-2*x)*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (5 x -\frac {3}{2} x^{2}-\frac {2}{3} x^{3}-\frac {5}{12} x^{4}-\frac {3}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-2 x +\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 55

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

\[ y(x)\to c_2 \left (-\frac {3 x^5}{10}-\frac {5 x^4}{12}-\frac {2 x^3}{3}-\frac {3 x^2}{2}+5 x+(1-2 x) \log (x)\right )+c_1 (1-2 x) \]

✓ Sympy. Time used: 0.376 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{5}}{450} + \frac {x^{4}}{36} + \frac {2 x^{3}}{9} + x^{2} + 2 x + 1\right ) + O\left (x^{6}\right ) \]