43.18.1 problem 1(a)

Internal problem ID [9004]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 159
Problem number : 1(a)
Date solved : Tuesday, September 30, 2025 at 06:01:12 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 3 x^{2} y^{\prime \prime }+5 x y^{\prime }+3 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 52

Order:=8; 
ode:=3*x^2*diff(diff(y(x),x),x)+5*x*diff(y(x),x)+3*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-3 x +\frac {9}{8} x^{2}-\frac {9}{56} x^{3}+\frac {27}{2240} x^{4}-\frac {81}{145600} x^{5}+\frac {81}{4659200} x^{6}-\frac {243}{619673600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{{2}/{3}}}+c_2 \left (1-\frac {3}{5} x +\frac {9}{80} x^{2}-\frac {9}{880} x^{3}+\frac {27}{49280} x^{4}-\frac {81}{4188800} x^{5}+\frac {81}{167552000} x^{6}-\frac {243}{26975872000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 111

ode=3*x^2*D[y[x],{x,2}]+5*x*D[y[x],x]+3*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {243 x^7}{26975872000}+\frac {81 x^6}{167552000}-\frac {81 x^5}{4188800}+\frac {27 x^4}{49280}-\frac {9 x^3}{880}+\frac {9 x^2}{80}-\frac {3 x}{5}+1\right )+\frac {c_2 \left (-\frac {243 x^7}{619673600}+\frac {81 x^6}{4659200}-\frac {81 x^5}{145600}+\frac {27 x^4}{2240}-\frac {9 x^3}{56}+\frac {9 x^2}{8}-3 x+1\right )}{x^{2/3}} \]

✓ Sympy. Time used: 0.391 (sec). Leaf size: 107

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*Derivative(y(x), (x, 2)) + 3*x*y(x) + 5*x*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {243 x^{7}}{26975872000} + \frac {81 x^{6}}{167552000} - \frac {81 x^{5}}{4188800} + \frac {27 x^{4}}{49280} - \frac {9 x^{3}}{880} + \frac {9 x^{2}}{80} - \frac {3 x}{5} + 1\right ) + \frac {C_{1} \left (- \frac {243 x^{7}}{619673600} + \frac {81 x^{6}}{4659200} - \frac {81 x^{5}}{145600} + \frac {27 x^{4}}{2240} - \frac {9 x^{3}}{56} + \frac {9 x^{2}}{8} - 3 x + 1\right )}{x^{\frac {2}{3}}} + O\left (x^{8}\right ) \]