46.2.14 problem 16

Internal problem ID [7317]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number : 16
Date solved : Wednesday, March 05, 2025 at 04:23:13 AM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\left (\frac {1}{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: 0.012 (sec). Leaf size: 36

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

\[ y = c_{1} \sqrt {x}\, \left (1+\frac {1}{2} x -\frac {1}{40} x^{2}-\frac {1}{560} x^{3}-\frac {1}{2688} x^{4}-\frac {1}{8448} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+4 x +\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 55

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^5}{8448}-\frac {x^4}{2688}-\frac {x^3}{560}-\frac {x^2}{40}+\frac {x}{2}+1\right )+c_2 (4 x+1) \]

✓ Sympy. Time used: 1.180 (sec). Leaf size: 73

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {128 x^{5}}{14175} + \frac {32 x^{4}}{315} - \frac {32 x^{3}}{45} + \frac {8 x^{2}}{3} - 4 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {32 x^{4}}{2835} - \frac {32 x^{3}}{315} + \frac {8 x^{2}}{15} - \frac {4 x}{3} + 1\right ) + O\left (x^{6}\right ) \]