28.5.3 problem 8

Internal problem ID [7206]
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 : 8
Date solved : Tuesday, September 30, 2025 at 04:25:28 PM
CAS classification : [_Jacobi]

\begin{align*} 2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 44

Order:=6; 
ode:=2*x*(1-x)*diff(diff(y(x),x),x)+(1-11*x)*diff(y(x),x)-10*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+5 x +14 x^{2}+30 x^{3}+55 x^{4}+91 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+10 x +35 x^{2}+84 x^{3}+165 x^{4}+286 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 65

ode=2*x*(1-x)*D[y[x],{x,2}]+(1-11*x)*D[y[x],x]-10*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (91 x^5+55 x^4+30 x^3+14 x^2+5 x+1\right )+c_2 \left (286 x^5+165 x^4+84 x^3+35 x^2+10 x+1\right ) \]

✓ Sympy. Time used: 0.467 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*(1 - x)*Derivative(y(x), (x, 2)) + (1 - 11*x)*Derivative(y(x), x) - 10*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {500 x^{5}}{567} + \frac {250 x^{4}}{63} - \frac {100 x^{3}}{9} + \frac {50 x^{2}}{3} - 10 x + 1\right ) + C_{1} \sqrt {x} \left (\frac {250 x^{4}}{567} - \frac {100 x^{3}}{63} + \frac {10 x^{2}}{3} - \frac {10 x}{3} + 1\right ) + O\left (x^{6}\right ) \]