78.21.2 problem 2 (b)

Internal problem ID [18367]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 31. Gauss Hypergeometric Equation. Problems at page 240
Problem number : 2 (b)
Date solved : Thursday, March 13, 2025 at 11:55:03 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (2 x^{2}+2 x \right ) y^{\prime \prime }+\left (1+5 x \right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 38

Order:=6; 
ode:=(2*x^2+2*x)*diff(diff(y(x),x),x)+(1+5*x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (-x^{5}+x^{4}-x^{3}+x^{2}-x +1\right ) \left (\sqrt {x}\, c_{1} +c_{2} \right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 57

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

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

✓ Sympy. Time used: 0.813 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((5*x + 1)*Derivative(y(x), x) + (2*x**2 + 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)
 

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