34.5.4 problem 9

Internal problem ID [6072]
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 : 9
Date solved : Sunday, March 30, 2025 at 10:37:47 AM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\frac {\left (1-2 x \right ) y^{\prime }}{3}+\frac {20 y}{9}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.039 (sec). Leaf size: 36

Order:=6; 
ode:=x*(1-x)*diff(diff(y(x),x),x)+1/3*(-2*x+1)*diff(y(x),x)+20/9*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{2}/{3}} \left (1-\frac {6}{5} x +\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-\frac {20}{3} x +\frac {35}{9} x^{2}+\frac {50}{81} x^{3}+\frac {65}{243} x^{4}+\frac {112}{729} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 57

ode=x*(1-x)*D[y[x],{x,2}]+1/3*(1-2*x)*D[y[x],x]+20/9*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (1-\frac {6 x}{5}\right ) x^{2/3}+c_2 \left (\frac {112 x^5}{729}+\frac {65 x^4}{243}+\frac {50 x^3}{81}+\frac {35 x^2}{9}-\frac {20 x}{3}+1\right ) \]

✓ Sympy. Time used: 1.107 (sec). Leaf size: 68

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x)*Derivative(y(x), (x, 2)) + (1 - 2*x)*Derivative(y(x), x)/3 + 20*y(x)/9,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {162000 x^{5}}{91} + \frac {13500 x^{4}}{7} + \frac {9000 x^{3}}{7} + 450 x^{2} + 60 x + 1\right ) + C_{1} x^{\frac {2}{3}} \left (\frac {6750 x^{4}}{77} + \frac {900 x^{3}}{11} + 45 x^{2} + 12 x + 1\right ) + O\left (x^{6}\right ) \]