12.16.4 problem Example 7.7.4 page 387

Internal problem ID [2066]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : Example 7.7.4 page 387
Date solved : Tuesday, March 04, 2025 at 01:49:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-13 x^{2}+7\right ) y^{\prime }-14 x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 32

Order:=6; 
ode:=x^2*(-2*x^2+1)*diff(diff(y(x),x),x)+x*(-13*x^2+7)*diff(y(x),x)-14*x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1+\frac {7}{8} x^{2}+\frac {77}{80} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-86400+216000 x^{2}-54000 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{6}} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 44

ode=x^2*(1-2*x^2)*D[y[x],{x,2}]+x*(7-13*x^2)*D[y[x],x]-14*x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {77 x^4}{80}+\frac {7 x^2}{8}+1\right )+c_1 \left (\frac {1}{x^6}-\frac {5}{2 x^4}+\frac {5}{8 x^2}\right ) \]

✓ Sympy. Time used: 1.314 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - 2*x**2)*Derivative(y(x), (x, 2)) - 14*x**2*y(x) + x*(7 - 13*x**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {2401 x^{5}}{29700} + \frac {343 x^{4}}{1080} - \frac {49 x^{3}}{54} + \frac {7 x^{2}}{4} - 2 x + 1\right ) + O\left (x^{6}\right ) \]