78.19.13 problem 4 (b)

Internal problem ID [18361]
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 29. Regular singular Points. Problems at page 227
Problem number : 4 (b)
Date solved : Monday, March 31, 2025 at 05:26:27 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 2 x y^{\prime \prime }+\left (3-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:=2*x*diff(diff(y(x),x),x)+(3-x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 \left (1+\frac {1}{3} x +\frac {1}{15} x^{2}+\frac {1}{105} x^{3}+\frac {1}{945} x^{4}+\frac {1}{10395} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 85

ode=2*x*D[y[x],{x,2}]+(3-x)*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^5}{10395}+\frac {x^4}{945}+\frac {x^3}{105}+\frac {x^2}{15}+\frac {x}{3}+1\right )+\frac {c_2 \left (\frac {x^5}{3840}+\frac {x^4}{384}+\frac {x^3}{48}+\frac {x^2}{8}+\frac {x}{2}+1\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 0.911 (sec). Leaf size: 65

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{5}}{10395} + \frac {x^{4}}{945} + \frac {x^{3}}{105} + \frac {x^{2}}{15} + \frac {x}{3} + 1\right ) + \frac {C_{1} \left (\frac {x^{5}}{3840} + \frac {x^{4}}{384} + \frac {x^{3}}{48} + \frac {x^{2}}{8} + \frac {x}{2} + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]