54.4.13 problem 13

Internal problem ID [8597]
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 : 13
Date solved : Sunday, March 30, 2025 at 01:20:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 52

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

\[ y = c_1 \sqrt {x}\, \left (1-\frac {5}{3} x +\frac {7}{6} x^{2}-\frac {1}{2} x^{3}+\frac {11}{72} x^{4}-\frac {13}{360} x^{5}+\frac {1}{144} x^{6}-\frac {17}{15120} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1-4 x +4 x^{2}-\frac {32}{15} x^{3}+\frac {16}{21} x^{4}-\frac {64}{315} x^{5}+\frac {64}{1485} x^{6}-\frac {1024}{135135} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 109

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {17 x^7}{15120}+\frac {x^6}{144}-\frac {13 x^5}{360}+\frac {11 x^4}{72}-\frac {x^3}{2}+\frac {7 x^2}{6}-\frac {5 x}{3}+1\right )+c_2 \left (-\frac {1024 x^7}{135135}+\frac {64 x^6}{1485}-\frac {64 x^5}{315}+\frac {16 x^4}{21}-\frac {32 x^3}{15}+4 x^2-4 x+1\right ) \]

✓ Sympy. Time used: 1.066 (sec). Leaf size: 95

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) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {1024 x^{7}}{135135} + \frac {64 x^{6}}{1485} - \frac {64 x^{5}}{315} + \frac {16 x^{4}}{21} - \frac {32 x^{3}}{15} + 4 x^{2} - 4 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{6}}{144} - \frac {13 x^{5}}{360} + \frac {11 x^{4}}{72} - \frac {x^{3}}{2} + \frac {7 x^{2}}{6} - \frac {5 x}{3} + 1\right ) + O\left (x^{8}\right ) \]