50.21.1 problem 2(a)

Internal problem ID [8138]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.6. Gauss Hypergeometric Equation. Page 187
Problem number : 2(a)
Date solved : Wednesday, March 05, 2025 at 05:30:38 AM
CAS classification : [_Jacobi]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 40

Order:=8; 
ode:=x*(1-x)*diff(diff(y(x),x),x)+(3/2-2*x)*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1-\frac {9}{2} x +\frac {15}{8} x^{2}+\frac {7}{16} x^{3}+\frac {27}{128} x^{4}+\frac {33}{256} x^{5}+\frac {91}{1024} x^{6}+\frac {135}{2048} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} \left (1-\frac {4}{3} x +\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 71

ode=x*(1-x)*D[y[x],{x,2}]+(3/2-2*x)*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {c_2 \left (\frac {135 x^7}{2048}+\frac {91 x^6}{1024}+\frac {33 x^5}{256}+\frac {27 x^4}{128}+\frac {7 x^3}{16}+\frac {15 x^2}{8}-\frac {9 x}{2}+1\right )}{\sqrt {x}}+c_1 \left (1-\frac {4 x}{3}\right ) \]

✓ Sympy. Time used: 1.112 (sec). Leaf size: 107

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x)*Derivative(y(x), (x, 2)) + (3/2 - 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=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {1024 x^{7}}{638512875} + \frac {256 x^{6}}{6081075} + \frac {128 x^{5}}{155925} + \frac {32 x^{4}}{2835} + \frac {32 x^{3}}{315} + \frac {8 x^{2}}{15} + \frac {4 x}{3} + 1\right ) + \frac {C_{1} \left (\frac {1024 x^{7}}{42567525} + \frac {256 x^{6}}{467775} + \frac {128 x^{5}}{14175} + \frac {32 x^{4}}{315} + \frac {32 x^{3}}{45} + \frac {8 x^{2}}{3} + 4 x + 1\right )}{\sqrt {x}} + O\left (x^{8}\right ) \]