12.16.13 problem 9

Internal problem ID [2075]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 9
Date solved : Saturday, March 29, 2025 at 11:47:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (3+8 x \right ) y^{\prime }-\left (5-49 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: 62

Order:=6; 
ode:=4*x^2*(1+x)*diff(diff(y(x),x),x)+4*x*(3+8*x)*diff(y(x),x)-(5-49*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{3} \left (1-4 x +10 x^{2}-20 x^{3}+35 x^{4}-56 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (\left (-36\right ) x^{3}+144 x^{4}-360 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+6 x +12 x^{2}-240 x^{3}+852 x^{4}-2022 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{{5}/{2}}} \]

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 86

ode=4*x^2*(1+x)*D[y[x],{x,2}]+4*x*(3+8*x)*D[y[x],x]-(5-49*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (35 x^{9/2}-20 x^{7/2}+10 x^{5/2}-4 x^{3/2}+\sqrt {x}\right )+c_1 \left (\frac {62 x^4-20 x^3+2 x^2+x+2}{2 x^{5/2}}+3 \sqrt {x} (4 x-1) \log (x)\right ) \]

✓ Sympy. Time used: 1.066 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*(x + 1)*Derivative(y(x), (x, 2)) + 4*x*(8*x + 3)*Derivative(y(x), x) - (5 - 49*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} + \frac {C_{1}}{x^{\frac {5}{2}}} + O\left (x^{6}\right ) \]