42.1.14 problem 3.24 (i)

Internal problem ID [6836]
Book : Advanced Mathematical Methods for Scientists and Engineers, Bender and Orszag. Springer October 29, 1999
Section : Chapter 3. APPROXIMATE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS. page 136
Problem number : 3.24 (i)
Date solved : Sunday, March 30, 2025 at 11:24:06 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 44

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

\[ y = c_1 \sqrt {x}\, \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+2 x +\frac {4}{3} x^{2}+\frac {8}{15} x^{3}+\frac {16}{105} x^{4}+\frac {32}{945} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 79

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 \left (\frac {32 x^5}{945}+\frac {16 x^4}{105}+\frac {8 x^3}{15}+\frac {4 x^2}{3}+2 x+1\right ) \]

✓ Sympy. Time used: 1.118 (sec). Leaf size: 65

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

\[ y{\left (x \right )} = C_{2} \left (\frac {32 x^{5}}{945} + \frac {16 x^{4}}{105} + \frac {8 x^{3}}{15} + \frac {4 x^{2}}{3} + 2 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4}}{24} + \frac {x^{3}}{6} + \frac {x^{2}}{2} + x + 1\right ) + O\left (x^{6}\right ) \]