54.4.8 problem 8

Internal problem ID [8592]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots Nonintegral. Exercises page 365
Problem number : 8
Date solved : Sunday, March 30, 2025 at 01:20:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 52

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+x +\frac {3}{8} x^{2}+\frac {1}{12} x^{3}+\frac {5}{384} x^{4}+\frac {1}{640} x^{5}+\frac {7}{46080} x^{6}+\frac {1}{80640} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {3}{2} x -\frac {13}{16} x^{2}-\frac {31}{144} x^{3}-\frac {173}{4608} x^{4}-\frac {187}{38400} x^{5}-\frac {463}{921600} x^{6}-\frac {971}{22579200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 151

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

\[ y(x)\to c_1 \left (\frac {x^7}{80640}+\frac {7 x^6}{46080}+\frac {x^5}{640}+\frac {5 x^4}{384}+\frac {x^3}{12}+\frac {3 x^2}{8}+x+1\right )+c_2 \left (-\frac {971 x^7}{22579200}-\frac {463 x^6}{921600}-\frac {187 x^5}{38400}-\frac {173 x^4}{4608}-\frac {31 x^3}{144}-\frac {13 x^2}{16}+\left (\frac {x^7}{80640}+\frac {7 x^6}{46080}+\frac {x^5}{640}+\frac {5 x^4}{384}+\frac {x^3}{12}+\frac {3 x^2}{8}+x+1\right ) \log (x)-\frac {3 x}{2}\right ) \]

✓ Sympy. Time used: 0.884 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{7}}{80640} + \frac {7 x^{6}}{46080} + \frac {x^{5}}{640} + \frac {5 x^{4}}{384} + \frac {x^{3}}{12} + \frac {3 x^{2}}{8} + x + 1\right ) + O\left (x^{8}\right ) \]