68.17.22 problem 22

Internal problem ID [17840]
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 : 22
Date solved : Thursday, October 02, 2025 at 02:28:46 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 38

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

\[ y = \left (3 x -3 x^{2}+\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-2 x +x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 59

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

\[ y(x)\to c_1 \left (x^2-2 x+1\right )+c_2 \left (\frac {x^5}{30}+\frac {x^4}{12}+\frac {x^3}{3}-3 x^2+\left (x^2-2 x+1\right ) \log (x)+3 x\right ) \]

✓ Sympy. Time used: 0.323 (sec). Leaf size: 32

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

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