54.4.16 problem 16

Internal problem ID [8600]
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 : 16
Date solved : Wednesday, March 05, 2025 at 06:10:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }+x \left (4 x -1\right ) y^{\prime }+2 \left (3 x -1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 55

Order:=8; 
ode:=2*x^2*diff(diff(y(x),x),x)+x*(4*x-1)*diff(y(x),x)+2*(3*x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1+\frac {4}{3} x +\frac {16}{3} x^{2}-\frac {64}{3} x^{3}+\frac {256}{9} x^{4}-\frac {1024}{45} x^{5}+\frac {4096}{315} x^{6}-\frac {16384}{2835} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} x^{2} \left (1-2 x +2 x^{2}-\frac {4}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{15} x^{5}+\frac {4}{45} x^{6}-\frac {8}{315} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 112

ode=2*x^2*D[y[x],{x,2}]+x*(4*x-1)*D[y[x],x]+2*(3*x-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {8 x^7}{315}+\frac {4 x^6}{45}-\frac {4 x^5}{15}+\frac {2 x^4}{3}-\frac {4 x^3}{3}+2 x^2-2 x+1\right ) x^2+\frac {c_2 \left (-\frac {16384 x^7}{2835}+\frac {4096 x^6}{315}-\frac {1024 x^5}{45}+\frac {256 x^4}{9}-\frac {64 x^3}{3}+\frac {16 x^2}{3}+\frac {4 x}{3}+1\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 0.991 (sec). Leaf size: 95

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) + x*(4*x - 1)*Derivative(y(x), x) + (6*x - 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} x^{2} \left (- \frac {4 x^{5}}{15} + \frac {2 x^{4}}{3} - \frac {4 x^{3}}{3} + 2 x^{2} - 2 x + 1\right ) + \frac {C_{1} \left (- \frac {16384 x^{7}}{2835} + \frac {4096 x^{6}}{315} - \frac {1024 x^{5}}{45} + \frac {256 x^{4}}{9} - \frac {64 x^{3}}{3} + \frac {16 x^{2}}{3} + \frac {4 x}{3} + 1\right )}{\sqrt {x}} + O\left (x^{8}\right ) \]