43.17.2 problem 1(b)

Internal problem ID [8995]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number : 1(b)
Date solved : Tuesday, September 30, 2025 at 06:01:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} y^{\prime \prime }+x^{6} y^{\prime }+2 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 70

Order:=8; 
ode:=3*x^2*diff(diff(y(x),x),x)+x^6*diff(y(x),x)+2*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 x \left (1-\frac {1}{3} x +\frac {1}{27} x^{2}-\frac {1}{486} x^{3}+\frac {1}{14580} x^{4}-\frac {7291}{656100} x^{5}+\frac {225991}{41334300} x^{6}-\frac {2522341}{3472081200} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-\frac {2}{3} x +\frac {2}{9} x^{2}-\frac {2}{81} x^{3}+\frac {1}{729} x^{4}-\frac {1}{21870} x^{5}+\frac {7291}{984150} x^{6}-\frac {225991}{62001450} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {1}{3} x^{2}+\frac {14}{243} x^{3}-\frac {35}{8748} x^{4}+\frac {101}{656100} x^{5}+\frac {69199}{14762250} x^{6}+\frac {19882543}{4340101500} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 121

ode=3*x^2*D[y[x],{x,2}]+x^6*D[y[x],x]+2*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x \left (7291 x^5-45 x^4+1350 x^3-24300 x^2+218700 x-656100\right ) \log (x)}{984150}+\frac {-80332 x^6+5895 x^5-158625 x^4+2430000 x^3-16402500 x^2+19683000 x+29524500}{29524500}\right )+c_2 \left (\frac {225991 x^7}{41334300}-\frac {7291 x^6}{656100}+\frac {x^5}{14580}-\frac {x^4}{486}+\frac {x^3}{27}-\frac {x^2}{3}+x\right ) \]

✓ Sympy. Time used: 0.362 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**6*Derivative(y(x), x) + 3*x**2*Derivative(y(x), (x, 2)) + 2*x*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {225991 x^{6}}{41334300} - \frac {7291 x^{5}}{656100} + \frac {x^{4}}{14580} - \frac {x^{3}}{486} + \frac {x^{2}}{27} - \frac {x}{3} + 1\right ) + O\left (x^{8}\right ) \]