23.5.2 problem 3(a)

Internal problem ID [4179]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(a)
Date solved : Sunday, March 30, 2025 at 02:41:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {\left (-3 x^{2}+x \right ) y^{\prime }}{2 x^{3}+2 x^{2}}+\frac {y}{2 x^{3}+2 x^{2}}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 40

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

\[ y = \left (-x^{5}+x^{4}-x^{3}+x^{2}-x +1\right ) \left (c_1 \sqrt {x}+c_2 x \right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 59

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

\[ \{y(x)\}\to c_1 x \left (-x^5+x^4-x^3+x^2-x+1\right )+c_2 \sqrt {x} \left (-x^5+x^4-x^3+x^2-x+1\right ) \]

✓ Sympy. Time used: 0.887 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = C_{2} x + C_{1} \sqrt {x} + O\left (x^{6}\right ) \]