Toronto Math Forum

MAT244--2018F => MAT244--Tests => Quiz-3 => Topic started by: Victor Ivrii on October 12, 2018, 06:10:07 PM

Title: Q3 TUT 5102
Post by: Victor Ivrii on October 12, 2018, 06:10:07 PM
Find the Wronskian of two solutions of the given differential equation without solving the equation.
$$
x^2y''+xy'+(x^2-\nu^2)y=0 \qquad\text{Bessel's equation}.
$$
Title: Re: Q3 TUT 5102
Post by: Yiting Zhang on October 12, 2018, 06:36:23 PM
$$y'' + \frac{x}{x^2}y' + \frac{x^2-v^2}{x^2}y = 0$$
$$W = ce^{-\int p(t)dt}$$
$$p(t) = \frac{1}{x}$$
$$-\int p(t)dt = -\ln{x}$$
$$W = ce^{-\ln{x}} = ce^{\ln{\frac{1}{x}}} = \frac{c}{x}$$