54.9.13 problem 13

Internal problem ID [8683]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 13
Date solved : Sunday, March 30, 2025 at 01:23:27 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 2 x y^{\prime \prime }+y^{\prime }+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)+diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1-\frac {1}{3} x +\frac {1}{30} x^{2}-\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}-\frac {1}{1247400} x^{5}+\frac {1}{97297200} x^{6}-\frac {1}{10216206000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1-x +\frac {1}{6} x^{2}-\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}-\frac {1}{113400} x^{5}+\frac {1}{7484400} x^{6}-\frac {1}{681080400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 111

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^7}{10216206000}+\frac {x^6}{97297200}-\frac {x^5}{1247400}+\frac {x^4}{22680}-\frac {x^3}{630}+\frac {x^2}{30}-\frac {x}{3}+1\right )+c_2 \left (-\frac {x^7}{681080400}+\frac {x^6}{7484400}-\frac {x^5}{113400}+\frac {x^4}{2520}-\frac {x^3}{90}+\frac {x^2}{6}-x+1\right ) \]

✓ Sympy. Time used: 0.892 (sec). Leaf size: 78

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{7}}{681080400} + \frac {x^{6}}{7484400} - \frac {x^{5}}{113400} + \frac {x^{4}}{2520} - \frac {x^{3}}{90} + \frac {x^{2}}{6} - x + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{6}}{97297200} - \frac {x^{5}}{1247400} + \frac {x^{4}}{22680} - \frac {x^{3}}{630} + \frac {x^{2}}{30} - \frac {x}{3} + 1\right ) + O\left (x^{8}\right ) \]