Toronto Math Forum
MAT2442018F => MAT244Tests => Quiz3 => Topic started 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.
$$
x^2y''+xy'+(x^2\nu^2)y=0 \qquad\text{Bessel's equation}.
$$

$$y'' + \frac{x}{x^2}y' + \frac{x^2v^2}{x^2}y = 0$$
$$W = ce^{\int p(t)dt}$$
$$p(t) = \frac{1}{x}$$
$$\int p(t)dt = \ln{x}$$
$$W = ce^{\ln{x}} = ce^{\ln{\frac{1}{x}}} = \frac{c}{x}$$