34.3.9 problem 10

Internal problem ID [6048]
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 : 10
Date solved : Wednesday, March 05, 2025 at 12:10:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (4 x^{3}-14 x^{2}-2 x \right ) y^{\prime \prime }-\left (6 x^{2}-7 x +1\right ) y^{\prime }+\left (6 x -1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 28

Order:=6; 
ode:=(4*x^3-14*x^2-2*x)*diff(diff(y(x),x),x)-(6*x^2-7*x+1)*diff(y(x),x)+(6*x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+2 x +\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-x +\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 25

ode=(4*x^3-14*x^2-2*x)*D[y[x],{x,2}]-(6*x^2-7*x+1)*D[y[x],x]+(6*x-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} (2 x+1)+c_2 (1-x) \]

✓ Sympy. Time used: 1.250 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((6*x - 1)*y(x) - (6*x**2 - 7*x + 1)*Derivative(y(x), x) + (4*x**3 - 14*x**2 - 2*x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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