34.5.2 problem 6

Internal problem ID [6070]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XVI. page 220
Problem number : 6
Date solved : Sunday, March 30, 2025 at 10:37:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x \left (1-x \right ) y^{\prime \prime }+x y^{\prime }-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:=2*x*(1-x)*diff(diff(y(x),x),x)+x*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +\left (1-\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{32} x^{3}+\frac {5}{384} x^{4}+\frac {7}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.052 (sec). Leaf size: 43

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

\[ y(x)\to c_1 \left (\frac {1}{384} \left (5 x^4+12 x^3+48 x^2-768 x+384\right )+\frac {1}{2} x \log (x)\right )+c_2 x \]

✓ Sympy. Time used: 0.943 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{120} + \frac {x^{3}}{24} + \frac {x^{2}}{6} + \frac {x}{2} + 1\right ) + C_{1} + O\left (x^{6}\right ) \]