12.14.22 problem 22

Internal problem ID [1963]
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 : 22
Date solved : Tuesday, March 04, 2025 at 01:47:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (4+x \right ) y^{\prime \prime }-x \left (1-3 x \right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 45

Order:=6; 
ode:=x^2*(x+4)*diff(diff(y(x),x),x)-x*(1-3*x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{4}} \left (1-\frac {9}{16} x +\frac {117}{512} x^{2}-\frac {663}{8192} x^{3}+\frac {13923}{524288} x^{4}-\frac {69615}{8388608} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (1-\frac {3}{7} x +\frac {12}{77} x^{2}-\frac {4}{77} x^{3}+\frac {24}{1463} x^{4}-\frac {24}{4807} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 86

ode=x^2*(4+x)*D[y[x],{x,2}]-x*(1-3*x)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x \left (-\frac {24 x^5}{4807}+\frac {24 x^4}{1463}-\frac {4 x^3}{77}+\frac {12 x^2}{77}-\frac {3 x}{7}+1\right )+c_2 \sqrt [4]{x} \left (-\frac {69615 x^5}{8388608}+\frac {13923 x^4}{524288}-\frac {663 x^3}{8192}+\frac {117 x^2}{512}-\frac {9 x}{16}+1\right ) \]

✓ Sympy. Time used: 0.992 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 4)*Derivative(y(x), (x, 2)) - x*(1 - 3*x)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x + C_{1} \sqrt [4]{x} + O\left (x^{6}\right ) \]