12.12.28 problem 34

Internal problem ID [1882]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 34
Date solved : Tuesday, March 04, 2025 at 01:45:36 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-2 x^{3}+1\right ) y^{\prime \prime }-10 x^{2} y^{\prime }-8 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

Order:=6; 
ode:=(-2*x^3+1)*diff(diff(y(x),x),x)-10*x^2*diff(y(x),x)-8*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {4 x^{3}}{3}\right ) y \left (0\right )+\left (x +\frac {3}{2} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 28

ode=(1-2*x^3)*D[y[x],{x,2}]-10*x^2*D[y[x],x]-8*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {3 x^4}{2}+x\right )+c_1 \left (\frac {4 x^3}{3}+1\right ) \]

✓ Sympy. Time used: 0.872 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-10*x**2*Derivative(y(x), x) - 8*x*y(x) + (1 - 2*x**3)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {4 x^{3}}{3} + 1\right ) + C_{1} x \left (\frac {3 x^{3}}{2} + 1\right ) + O\left (x^{6}\right ) \]