64.14.14 problem 14

Internal problem ID [13502]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.1. Exercises page 232
Problem number : 14
Date solved : Monday, March 31, 2025 at 07:59:47 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=5 \end{align*}

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

Order:=6; 
ode:=(2*x^2-3)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+y(x) = 0; 
ic:=y(0) = -1, D(y)(0) = 5; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = -1+5 x -\frac {1}{6} x^{2}-\frac {5}{18} x^{3}-\frac {1}{216} x^{4}-\frac {7}{216} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 36

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

\[ y(x)\to -\frac {7 x^5}{216}-\frac {x^4}{216}-\frac {5 x^3}{18}-\frac {x^2}{6}+5 x-1 \]

✓ Sympy. Time used: 0.827 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*Derivative(y(x), x) + (2*x**2 - 3)*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 5} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{216} + \frac {x^{2}}{6} + 1\right ) + C_{1} x \left (1 - \frac {x^{2}}{18}\right ) + O\left (x^{6}\right ) \]