56.4.31 problem 27

Internal problem ID [8920]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 27
Date solved : Wednesday, March 05, 2025 at 07:08:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 35

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

\[ y = c_{1} \sqrt {x}\, \left (1+\frac {15}{8} x^{2}+\frac {189}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1+\frac {6}{7} x^{2}+\frac {45}{77} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 50

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

\[ y(x)\to c_1 \left (\frac {45 x^4}{77}+\frac {6 x^2}{7}+1\right ) x^2+c_2 \left (\frac {189 x^4}{128}+\frac {15 x^2}{8}+1\right ) \sqrt {x} \]

✓ Sympy. Time used: 1.146 (sec). Leaf size: 17

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

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