6.16.19 problem 15

Internal problem ID [2081]
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 : 15
Date solved : Tuesday, September 30, 2025 at 05:23:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (1+2 x \right ) y^{\prime \prime }-2 x \left (4-x \right ) y^{\prime }-\left (7+5 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 42

Order:=6; 
ode:=4*x^2*(2*x+1)*diff(diff(y(x),x),x)-2*x*(4-x)*diff(y(x),x)-(7+5*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{4} \left (1-\frac {18}{5} x +\frac {39}{4} x^{2}-\frac {663}{28} x^{3}+\frac {13923}{256} x^{4}-\frac {7735}{64} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-144-\frac {405}{8} x^{4}+\frac {729}{4} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 67

ode=4*x^2*(1+2*x)*D[y[x],{x,2}]-2*x*(4-x)*D[y[x],x]-(7+5*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{\sqrt {x}}-\frac {35 x^{7/2}}{128}\right )+c_2 \left (\frac {13923 x^{15/2}}{256}-\frac {663 x^{13/2}}{28}+\frac {39 x^{11/2}}{4}-\frac {18 x^{9/2}}{5}+x^{7/2}\right ) \]

✓ Sympy. Time used: 0.461 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*(2*x + 1)*Derivative(y(x), (x, 2)) - 2*x*(4 - x)*Derivative(y(x), x) - (5*x + 7)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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