54.4.14 problem 14

Internal problem ID [8598]
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 : 14
Date solved : Wednesday, March 05, 2025 at 06:10:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x y^{\prime \prime }+\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: 42

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

\[ y = c_{1} \sqrt {x}\, \left (1+\frac {4}{3} x +\frac {4}{15} x^{2}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+5 x +\frac {5}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{168} x^{4}+\frac {1}{2520} x^{5}-\frac {1}{33264} x^{6}+\frac {1}{432432} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 76

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {4 x^2}{15}+\frac {4 x}{3}+1\right )+c_2 \left (\frac {x^7}{432432}-\frac {x^6}{33264}+\frac {x^5}{2520}-\frac {x^4}{168}+\frac {x^3}{6}+\frac {5 x^2}{2}+5 x+1\right ) \]

✓ Sympy. Time used: 0.947 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), (x, 2)) + (2*x + 1)*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)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{7}}{432432} - \frac {x^{6}}{33264} + \frac {x^{5}}{2520} - \frac {x^{4}}{168} + \frac {x^{3}}{6} + \frac {5 x^{2}}{2} + 5 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {4 x^{2}}{15} + \frac {4 x}{3} + 1\right ) + O\left (x^{8}\right ) \]