54.8.8 problem 10

Internal problem ID [8670]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises page 391
Problem number : 10
Date solved : Sunday, March 30, 2025 at 01:23:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 52

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

\[ y = c_1 \sqrt {x}\, \left (1+\frac {1}{2} x +\frac {9}{40} x^{2}+\frac {103}{1680} x^{3}+\frac {187}{13440} x^{4}+\frac {247}{98560} x^{5}+\frac {17861}{46126080} x^{6}+\frac {23767}{461260800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 75

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

\[ y(x)\to c_2 \left (-\frac {x^7}{5376}-\frac {x^6}{1920}-\frac {x^5}{640}-\frac {x^4}{192}-\frac {x^3}{48}-\frac {x^2}{8}+\frac {x}{2}+\left (1-\frac {x}{2}\right ) \log (x)\right )+c_1 \left (1-\frac {x}{2}\right ) \]

✓ Sympy. Time used: 1.001 (sec). Leaf size: 87

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), (x, 2)) + (1 - x)*Derivative(y(x), x) - (x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{7}}{5040} + \frac {x^{6}}{720} + \frac {x^{5}}{120} + \frac {x^{4}}{24} + \frac {x^{3}}{6} + \frac {x^{2}}{2} + x + 1\right ) + C_{1} \sqrt {x} \left (\frac {17861 x^{6}}{46126080} + \frac {247 x^{5}}{98560} + \frac {187 x^{4}}{13440} + \frac {103 x^{3}}{1680} + \frac {9 x^{2}}{40} + \frac {x}{2} + 1\right ) + O\left (x^{8}\right ) \]