50.19.15 problem 4(b)

Internal problem ID [8123]
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 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 4(b)
Date solved : Sunday, March 30, 2025 at 12:46:01 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: 0.031 (sec). Leaf size: 52

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

\[ y = \frac {c_1 \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\frac {1}{46080} x^{6}+\frac {1}{645120} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_2 \left (1+\frac {1}{3} x +\frac {1}{15} x^{2}+\frac {1}{105} x^{3}+\frac {1}{945} x^{4}+\frac {1}{10395} x^{5}+\frac {1}{135135} x^{6}+\frac {1}{2027025} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 113

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

\[ y(x)\to c_1 \left (\frac {x^7}{2027025}+\frac {x^6}{135135}+\frac {x^5}{10395}+\frac {x^4}{945}+\frac {x^3}{105}+\frac {x^2}{15}+\frac {x}{3}+1\right )+\frac {c_2 \left (\frac {x^7}{645120}+\frac {x^6}{46080}+\frac {x^5}{3840}+\frac {x^4}{384}+\frac {x^3}{48}+\frac {x^2}{8}+\frac {x}{2}+1\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 0.974 (sec). Leaf size: 85

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=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{7}}{2027025} + \frac {x^{6}}{135135} + \frac {x^{5}}{10395} + \frac {x^{4}}{945} + \frac {x^{3}}{105} + \frac {x^{2}}{15} + \frac {x}{3} + 1\right ) + \frac {C_{1} \left (\frac {x^{7}}{645120} + \frac {x^{6}}{46080} + \frac {x^{5}}{3840} + \frac {x^{4}}{384} + \frac {x^{3}}{48} + \frac {x^{2}}{8} + \frac {x}{2} + 1\right )}{\sqrt {x}} + O\left (x^{8}\right ) \]