74.17.21 problem 21

Internal problem ID [16473]
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 : 21
Date solved : Thursday, March 13, 2025 at 08:14:30 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 44

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

\[ y = c_{1} \sqrt {x}\, \left (1+\frac {3}{2} x +\frac {15}{8} x^{2}+\frac {35}{16} x^{3}+\frac {315}{128} x^{4}+\frac {693}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+2 x +\frac {8}{3} x^{2}+\frac {16}{5} x^{3}+\frac {128}{35} x^{4}+\frac {256}{63} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 83

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {693 x^5}{256}+\frac {315 x^4}{128}+\frac {35 x^3}{16}+\frac {15 x^2}{8}+\frac {3 x}{2}+1\right )+c_2 \left (\frac {256 x^5}{63}+\frac {128 x^4}{35}+\frac {16 x^3}{5}+\frac {8 x^2}{3}+2 x+1\right ) \]

✓ Sympy. Time used: 0.980 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x)*Derivative(y(x), (x, 2)) + (1/2 - 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)
 

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