6.16.25 problem 21

Internal problem ID [2087]
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 : 21
Date solved : Tuesday, September 30, 2025 at 05:23:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (1+4 x \right ) y^{\prime }-\left (49+27 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 48

Order:=6; 
ode:=4*x^2*(1+x)*diff(diff(y(x),x),x)+4*x*(1+4*x)*diff(y(x),x)-(49+27*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{7} \left (1-2 x +3 x^{2}-4 x^{3}+5 x^{4}-6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (3628800-3024000 x +2419200 x^{2}-1814400 x^{3}+1209600 x^{4}-604800 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{7}/{2}}} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 86

ode=4*x^2*(1+x)*D[y[x],{x,2}]+4*x*(1+4*x)*D[y[x],x]-(49+27*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {2}{3 x^{3/2}}-\frac {5}{6 x^{5/2}}+\frac {1}{x^{7/2}}+\frac {\sqrt {x}}{3}-\frac {1}{2 \sqrt {x}}\right )+c_2 \left (5 x^{15/2}-4 x^{13/2}+3 x^{11/2}-2 x^{9/2}+x^{7/2}\right ) \]

✓ Sympy. Time used: 0.448 (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*(4*x + 1)*Derivative(y(x), x) - (27*x + 49)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {7}{2}} + \frac {C_{1}}{x^{\frac {7}{2}}} + O\left (x^{6}\right ) \]