54.4.11 problem 11

Internal problem ID [8595]
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 : 11
Date solved : Wednesday, March 05, 2025 at 06:10:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 42

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

\[ y = c_{1} \sqrt {x}\, \left (1-\frac {5}{8} x +\frac {7}{128} x^{2}+\frac {3}{1024} x^{3}+\frac {11}{32768} x^{4}+\frac {13}{262144} x^{5}+\frac {35}{4194304} x^{6}+\frac {51}{33554432} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {1}{2} x^{2}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 76

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

\[ y(x)\to c_2 \left (\frac {x^2}{2}-2 x+1\right )+c_1 \sqrt {x} \left (\frac {51 x^7}{33554432}+\frac {35 x^6}{4194304}+\frac {13 x^5}{262144}+\frac {11 x^4}{32768}+\frac {3 x^3}{1024}+\frac {7 x^2}{128}-\frac {5 x}{8}+1\right ) \]

✓ Sympy. Time used: 1.201 (sec). Leaf size: 100

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

\[ y{\left (x \right )} = C_{2} \left (\frac {131072 x^{7}}{42567525} + \frac {16384 x^{6}}{467775} + \frac {4096 x^{5}}{14175} + \frac {512 x^{4}}{315} + \frac {256 x^{3}}{45} + \frac {32 x^{2}}{3} + 8 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {16384 x^{6}}{6081075} + \frac {4096 x^{5}}{155925} + \frac {512 x^{4}}{2835} + \frac {256 x^{3}}{315} + \frac {32 x^{2}}{15} + \frac {8 x}{3} + 1\right ) + O\left (x^{8}\right ) \]