54.3.10 problem 10

Internal problem ID [8566]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 10
Date solved : Wednesday, March 05, 2025 at 06:09:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 x y^{\prime }+5 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 49

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

\[ y = \left (1-\frac {5}{2} x^{2}+\frac {15}{8} x^{4}-\frac {13}{16} x^{6}\right ) y \left (0\right )+\left (x -\frac {7}{6} x^{3}+\frac {77}{120} x^{5}-\frac {11}{48} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 56

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

\[ y(x)\to c_2 \left (-\frac {11 x^7}{48}+\frac {77 x^5}{120}-\frac {7 x^3}{6}+x\right )+c_1 \left (-\frac {13 x^6}{16}+\frac {15 x^4}{8}-\frac {5 x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.804 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), x) + 5*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {13 x^{6}}{16} + \frac {15 x^{4}}{8} - \frac {5 x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {77 x^{4}}{120} - \frac {7 x^{2}}{6} + 1\right ) + O\left (x^{8}\right ) \]