6.15.61 problem 62

Internal problem ID [2059]
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 II. Exercises 7.6. Page 374
Problem number : 62
Date solved : Tuesday, September 30, 2025 at 05:23:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 42

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

\[ y = \left (1-\frac {3}{4} x +\frac {9}{16} x^{2}-\frac {27}{64} x^{3}+\frac {81}{256} x^{4}-\frac {243}{1024} x^{5}\right ) x \left (\ln \left (x \right ) c_2 +c_1 \right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 84

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

\[ y(x)\to c_1 x \left (-\frac {243 x^5}{1024}+\frac {81 x^4}{256}-\frac {27 x^3}{64}+\frac {9 x^2}{16}-\frac {3 x}{4}+1\right )+c_2 x \left (-\frac {243 x^5}{1024}+\frac {81 x^4}{256}-\frac {27 x^3}{64}+\frac {9 x^2}{16}-\frac {3 x}{4}+1\right ) \log (x) \]

✓ Sympy. Time used: 0.307 (sec). Leaf size: 8

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

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