54.4.6 problem 6

Internal problem ID [8590]
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 : 6
Date solved : Wednesday, March 05, 2025 at 06:10:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x y^{\prime \prime }+5 \left (1-2 x \right ) y^{\prime }-5 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 40

Order:=8; 
ode:=2*x*diff(diff(y(x),x),x)+5*(-2*x+1)*diff(y(x),x)-5*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1+10 x +\operatorname {O}\left (x^{8}\right )\right )}{x^{{3}/{2}}}+c_{2} \left (1+x +\frac {15}{14} x^{2}+\frac {125}{126} x^{3}+\frac {625}{792} x^{4}+\frac {625}{1144} x^{5}+\frac {625}{1872} x^{6}+\frac {3125}{17136} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 65

ode=2*x*D[y[x],{x,2}]+5*(1-2*x)*D[y[x],x]-5*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {c_2 (10 x+1)}{x^{3/2}}+c_1 \left (\frac {3125 x^7}{17136}+\frac {625 x^6}{1872}+\frac {625 x^5}{1144}+\frac {625 x^4}{792}+\frac {125 x^3}{126}+\frac {15 x^2}{14}+x+1\right ) \]

✓ Sympy. Time used: 0.898 (sec). Leaf size: 63

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), (x, 2)) + (5 - 10*x)*Derivative(y(x), x) - 5*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {3125 x^{7}}{17136} + \frac {625 x^{6}}{1872} + \frac {625 x^{5}}{1144} + \frac {625 x^{4}}{792} + \frac {125 x^{3}}{126} + \frac {15 x^{2}}{14} + x + 1\right ) + \frac {C_{1} \left (10 x + 1\right )}{x^{\frac {3}{2}}} + O\left (x^{8}\right ) \]