54.7.3 problem 3

Internal problem ID [8651]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a Positive Integer: Logarithmic Case. Exercises page 384
Problem number : 3
Date solved : Sunday, March 30, 2025 at 01:22:23 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 2 x y^{\prime \prime }+6 y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 74

Order:=8; 
ode:=2*x*diff(diff(y(x),x),x)+6*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{6} x +\frac {1}{96} x^{2}-\frac {1}{2880} x^{3}+\frac {1}{138240} x^{4}-\frac {1}{9676800} x^{5}+\frac {1}{928972800} x^{6}-\frac {1}{117050572800} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) x^{2}+c_2 \left (\ln \left (x \right ) \left (\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{384} x^{4}-\frac {1}{11520} x^{5}+\frac {1}{552960} x^{6}-\frac {1}{38707200} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-x +\frac {1}{18} x^{3}-\frac {25}{4608} x^{4}+\frac {157}{691200} x^{5}-\frac {91}{16588800} x^{6}+\frac {709}{8128512000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 114

ode=2*x*D[y[x],{x,2}]+6*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (\frac {x^6}{928972800}-\frac {x^5}{9676800}+\frac {x^4}{138240}-\frac {x^3}{2880}+\frac {x^2}{96}-\frac {x}{6}+1\right )+c_1 \left (\frac {53 x^6-2244 x^5+55800 x^4-633600 x^3+1036800 x^2+8294400 x+16588800}{16588800 x^2}-\frac {\left (x^4-48 x^3+1440 x^2-23040 x+138240\right ) \log (x)}{1105920}\right ) \]

✓ Sympy. Time used: 0.751 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), (x, 2)) + y(x) + 6*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {x^{7}}{117050572800} + \frac {x^{6}}{928972800} - \frac {x^{5}}{9676800} + \frac {x^{4}}{138240} - \frac {x^{3}}{2880} + \frac {x^{2}}{96} - \frac {x}{6} + 1\right ) + O\left (x^{8}\right ) \]