87.26.18 problem 27

Internal problem ID [23852]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 253
Problem number : 27
Date solved : Thursday, October 02, 2025 at 09:45:51 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\left (\frac {2}{3}-3 x \right ) y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 44

Order:=6; 
ode:=x*(1-x)*diff(diff(y(x),x),x)+(2/3-3*x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{3}} \left (1+\frac {4}{3} x +\frac {14}{9} x^{2}+\frac {140}{81} x^{3}+\frac {455}{243} x^{4}+\frac {1456}{729} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {3}{2} x +\frac {9}{5} x^{2}+\frac {81}{40} x^{3}+\frac {243}{110} x^{4}+\frac {729}{308} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 85

ode=x*(1-x)*D[y[x],{x,2}]+(2/3-3*x)*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {1456 x^5}{729}+\frac {455 x^4}{243}+\frac {140 x^3}{81}+\frac {14 x^2}{9}+\frac {4 x}{3}+1\right )+c_2 \left (\frac {729 x^5}{308}+\frac {243 x^4}{110}+\frac {81 x^3}{40}+\frac {9 x^2}{5}+\frac {3 x}{2}+1\right ) \]

✓ Sympy. Time used: 0.394 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x)*Derivative(y(x), (x, 2)) + (2/3 - 3*x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [3]{x} + C_{1} + O\left (x^{6}\right ) \]