68.18.59 problem 65

Internal problem ID [17903]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Chapter 4 review exercises, page 219
Problem number : 65
Date solved : Thursday, October 02, 2025 at 02:29:27 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*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 39

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

\[ y = \left (1-\frac {3}{2} x^{2}-\frac {5}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 42

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

\[ y(x)\to c_2 \left (-\frac {x^5}{8}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {5 x^4}{8}-\frac {3 x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.287 (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 = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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