12.14.12 problem 9

Internal problem ID [1953]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 9
Date solved : Tuesday, March 04, 2025 at 01:46:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x^{2}+x +3\right ) y^{\prime \prime }+\left (-x^{2}+x +4\right ) y^{\prime }+x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 42

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

\[ y = \frac {c_1 \left (1-\frac {1}{18} x -\frac {71}{405} x^{2}+\frac {719}{34992} x^{3}-\frac {1678}{1082565} x^{4}-\frac {513547}{992023200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}}+c_2 \left (1-\frac {1}{14} x^{2}+\frac {1}{105} x^{3}-\frac {1}{3640} x^{4}-\frac {23}{54600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 80

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

\[ y(x)\to c_1 \left (-\frac {23 x^5}{54600}-\frac {x^4}{3640}+\frac {x^3}{105}-\frac {x^2}{14}+1\right )+\frac {c_2 \left (-\frac {513547 x^5}{992023200}-\frac {1678 x^4}{1082565}+\frac {719 x^3}{34992}-\frac {71 x^2}{405}-\frac {x}{18}+1\right )}{\sqrt [3]{x}} \]

✓ Sympy. Time used: 1.072 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**2 + x + 3)*Derivative(y(x), (x, 2)) + x*y(x) + (-x**2 + x + 4)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{2}}{\sqrt [3]{x}} + C_{1} + O\left (x^{6}\right ) \]