12.14.3 problem Example 7.5.3 page 356

Internal problem ID [1944]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : Example 7.5.3 page 356
Date solved : Tuesday, March 04, 2025 at 01:46:38 PM
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+2 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 47

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)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \sqrt {x}\, \left (1+2 x -\frac {9}{8} x^{2}+\frac {7}{4} x^{3}-\frac {607}{640} x^{4}+\frac {13347}{11200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{2} \left (1-\frac {2}{5} x +\frac {27}{35} x^{2}-\frac {34}{105} x^{3}+\frac {584}{1155} x^{4}-\frac {768}{3575} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 86

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

\[ y(x)\to c_1 \left (-\frac {768 x^5}{3575}+\frac {584 x^4}{1155}-\frac {34 x^3}{105}+\frac {27 x^2}{35}-\frac {2 x}{5}+1\right ) x^2+c_2 \left (\frac {13347 x^5}{11200}-\frac {607 x^4}{640}+\frac {7 x^3}{4}-\frac {9 x^2}{8}+2 x+1\right ) \sqrt {x} \]

✓ Sympy. Time used: 1.189 (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*x + 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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