74.16.16 problem 16

Internal problem ID [16451]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 16
Date solved : Monday, March 31, 2025 at 02:53:40 PM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

\[ y = -2+2 x +3 x^{2}-\frac {1}{3} x^{3}+\frac {5}{4} x^{4}-\frac {1}{4} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 34

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

\[ y(x)\to -\frac {x^5}{4}+\frac {5 x^4}{4}-\frac {x^3}{3}+3 x^2+2 x-2 \]

✓ Sympy. Time used: 0.789 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {5 x^{4}}{8} - \frac {3 x^{2}}{2} + 1\right ) + C_{1} x \left (1 - \frac {x^{2}}{6}\right ) + O\left (x^{6}\right ) \]