12.14.27 problem 29

Internal problem ID [1968]
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 : 29
Date solved : Saturday, March 29, 2025 at 11:44:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (3+4 x \right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }-\left (3+4 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 41

Order:=6; 
ode:=x^2*(3+4*x)*diff(diff(y(x),x),x)+x*(11+4*x)*diff(y(x),x)-(3+4*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+\frac {32}{7} x +\frac {48}{7} x^{2}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}}+c_2 \,x^{{1}/{3}} \left (1+\frac {32}{117} x -\frac {28}{1053} x^{2}+\frac {4480}{540189} x^{3}-\frac {15680}{4113747} x^{4}+\frac {401408}{185118615} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 67

ode=x^2*(3+4*x)*D[y[x],{x,2}]+x*(11+4*x)*D[y[x],x]-(3+4*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_2 \left (\frac {48 x^2}{7}+\frac {32 x}{7}+1\right )}{x^3}+c_1 \sqrt [3]{x} \left (\frac {401408 x^5}{185118615}-\frac {15680 x^4}{4113747}+\frac {4480 x^3}{540189}-\frac {28 x^2}{1053}+\frac {32 x}{117}+1\right ) \]

✓ Sympy. Time used: 0.973 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(4*x + 3)*Derivative(y(x), (x, 2)) + x*(4*x + 11)*Derivative(y(x), x) - (4*x + 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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