28.3.12 problem 14

Internal problem ID [7186]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XIV. page 177
Problem number : 14
Date solved : Tuesday, September 30, 2025 at 04:25:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (\left (1-n \right ) x -\left (6-4 n \right ) x^{2}\right ) y^{\prime }+n \left (1-n \right ) x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 122

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

\[ y = c_1 \,x^{n} \left (1+n x +\frac {1}{2} n \left (n +3\right ) x^{2}+\frac {1}{6} \left (n +5\right ) \left (n +4\right ) n x^{3}+\frac {1}{24} n \left (n +5\right ) \left (n +7\right ) \left (n +6\right ) x^{4}+\frac {1}{120} \left (n +9\right ) \left (n +8\right ) \left (n +7\right ) \left (n +6\right ) n x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-n x +\frac {1}{2} n \left (n -3\right ) x^{2}-\frac {1}{6} \left (n -4\right ) \left (n -5\right ) n x^{3}+\frac {1}{24} n \left (n -5\right ) \left (n -6\right ) \left (n -7\right ) x^{4}-\frac {1}{120} \left (n -6\right ) \left (n -7\right ) \left (n -8\right ) \left (n -9\right ) n x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 1878

ode=x^2*(1-4*x)*D[y[x],{x,2}]+((1-n)*x-(6-4*n)*x^2)*D[y[x],x]+n*(1-n)*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 0.754 (sec). Leaf size: 7

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

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