MAT244--2018F > Quiz-3

Q3 TUT 5101

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Victor Ivrii:
Find the Wronskian of two solutions of the given differential equation without solving the equation.
$$(1-x^2)y''-2xy'+\alpha(\alpha+1)y=0, \qquad\text{Legendre's equation}.$$

Yiran Zhu:
This is the hand written solution. I am still trying to figure out how to convert latex into plain text.

Yiting Zhang:
$$y′′−\frac{2x}{1−x^2}y′+\frac{\alpha(\alpha+1)}{1−x^2}y=0$$
$$W=ce^{\int −p(x) dx}, p(x) = −\frac{2x}{1−x^2}$$
$$W = ce^{\int \frac{2x}{1−x^2} dx}$$
$$u=1−x^2, du=−2xdx$$
$$ce^{-\int \frac{1}{u}du} = ce^{-ln(u)+C}=ce^{−ln(1−x^2)+C}=\frac{ce^C}{1-x^2}$$
Since $ce^C$ is constant. Let $ce^C = c$
$$W=\frac{c}{1−x^2}$$

Victor Ivrii:
Yiran, you use LaTeX math snippets on the forum
Yiting: ln must be escaped as \ln