15.24.6 problem 6

Internal problem ID [3378]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 42, page 206
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 04:37:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }-5 x y^{\prime }+9 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 36

Order:=6; 
ode:=x^2*(-x^2+1)*diff(diff(y(x),x),x)-5*x*diff(y(x),x)+9*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = x^{3} \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {3}{2} x^{2}+\frac {15}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {13}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 71

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

\[ y(x)\to c_1 \left (\frac {15 x^4}{8}+\frac {3 x^2}{2}+1\right ) x^3+c_2 \left (\left (-\frac {13 x^4}{32}-\frac {x^2}{4}\right ) x^3+\left (\frac {15 x^4}{8}+\frac {3 x^2}{2}+1\right ) x^3 \log (x)\right ) \]

✓ Sympy. Time used: 0.938 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - x**2)*Derivative(y(x), (x, 2)) - 5*x*Derivative(y(x), x) + 9*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{3} + O\left (x^{6}\right ) \]