45.2.1 problem 1

Internal problem ID [7224]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.2 page 239
Problem number : 1
Date solved : Sunday, March 30, 2025 at 11:51:42 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{3} y^{\prime \prime }+4 x^{2} y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=x^3*diff(diff(y(x),x),x)+4*x^2*diff(y(x),x)+3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.073 (sec). Leaf size: 282

ode=x^3*D[y[x],{x,2}]+4*x^2*D[y[x],x]+3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 e^{-\frac {2 i \sqrt {3}}{\sqrt {x}}} \left (-\frac {14315125825 i x^{9/2}}{8796093022208 \sqrt {3}}+\frac {8083075 i x^{7/2}}{4294967296 \sqrt {3}}-\frac {15015 i \sqrt {3} x^{5/2}}{8388608}+\frac {385 i \sqrt {3} x^{3/2}}{8192}+\frac {930483178625 x^5}{844424930131968}-\frac {509233725 x^4}{549755813888}+\frac {425425 x^3}{268435456}-\frac {5005 x^2}{524288}-\frac {315 x}{512}-\frac {35 i \sqrt {x}}{16 \sqrt {3}}+1\right )}{x^{5/4}}+\frac {c_2 e^{\frac {2 i \sqrt {3}}{\sqrt {x}}} \left (\frac {14315125825 i x^{9/2}}{8796093022208 \sqrt {3}}-\frac {8083075 i x^{7/2}}{4294967296 \sqrt {3}}+\frac {15015 i \sqrt {3} x^{5/2}}{8388608}-\frac {385 i \sqrt {3} x^{3/2}}{8192}+\frac {930483178625 x^5}{844424930131968}-\frac {509233725 x^4}{549755813888}+\frac {425425 x^3}{268435456}-\frac {5005 x^2}{524288}-\frac {315 x}{512}+\frac {35 i \sqrt {x}}{16 \sqrt {3}}+1\right )}{x^{5/4}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 2)) + 4*x**2*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)
 

ValueError : ODE x**3*Derivative(y(x), (x, 2)) + 4*x**2*Derivative(y(x), x) + 3*y(x) does not match hint 2nd_power_series_regular