15.23.13 problem 17

Internal problem ID [3363]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 17
Date solved : Tuesday, March 04, 2025 at 04:36:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (1-x \right ) y^{\prime \prime }+3 x \left (1+2 x \right ) y^{\prime }-3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 45

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

\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {13}{4} x +\frac {117}{32} x^{2}-\frac {195}{128} x^{3}+\frac {195}{2048} x^{4}+\frac {117}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{3}/{4}}}+c_{2} x \left (1-\frac {6}{11} x +\frac {4}{55} x^{2}+\frac {8}{1045} x^{3}+\frac {48}{24035} x^{4}+\frac {32}{43263} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 86

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

\[ y(x)\to c_1 x \left (\frac {32 x^5}{43263}+\frac {48 x^4}{24035}+\frac {8 x^3}{1045}+\frac {4 x^2}{55}-\frac {6 x}{11}+1\right )+\frac {c_2 \left (\frac {117 x^5}{8192}+\frac {195 x^4}{2048}-\frac {195 x^3}{128}+\frac {117 x^2}{32}-\frac {13 x}{4}+1\right )}{x^{3/4}} \]

✓ Sympy. Time used: 1.073 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = C_{2} x + \frac {C_{1}}{x^{\frac {3}{4}}} + O\left (x^{6}\right ) \]