7.16.3 problem 3

Internal problem ID [500]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.4 (Method of Frobenius: The exceptional cases). Problems at page 246
Problem number : 3
Date solved : Tuesday, March 04, 2025 at 11:25:27 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x y^{\prime \prime }+\left (5+3 x \right ) y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 44

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+(5+3*x)*diff(y(x),x)+3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {3}{5} x +\frac {3}{10} x^{2}-\frac {9}{70} x^{3}+\frac {27}{560} x^{4}-\frac {9}{560} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-144+432 x -648 x^{2}+648 x^{3}-486 x^{4}+\frac {1458}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{4}} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 64

ode=x*D[y[x],{x,2}]+(5+3*x)*D[y[x],x]+3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^4}-\frac {3}{x^3}+\frac {9}{2 x^2}-\frac {9}{2 x}+\frac {27}{8}\right )+c_2 \left (\frac {27 x^4}{560}-\frac {9 x^3}{70}+\frac {3 x^2}{10}-\frac {3 x}{5}+1\right ) \]

✓ Sympy. Time used: 0.936 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (3*x + 5)*Derivative(y(x), x) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {9 x^{5}}{560} + \frac {27 x^{4}}{560} - \frac {9 x^{3}}{70} + \frac {3 x^{2}}{10} - \frac {3 x}{5} + 1\right ) + \frac {C_{1} \left (- \frac {9 x^{3}}{2} + \frac {9 x^{2}}{2} - 3 x + 1\right )}{x^{4}} + O\left (x^{6}\right ) \]