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Friday, October 29, 2010

Bernouilli equations

I watched approximately half of video lecture 4 of 18.03 today. Topic of lecture 4 is solving differential equations by direct or indirect substitution.

Direct and indirect substitution originates from Calculus Integration, I suppose. Take for example, the integral:
$$\int{x \sin(x^2)\ dx}$$
requires the direct substitutions $y=x^2, dy=2x \ dx$ to solve. However, the integral
$$\int{\frac{1}{\sqrt{1-x^2}}\ dx}$$
requires the indirect substitution $x=sin(u), dx=cos(u) \ du$ to solve.

A Bernoulli equation is a DE of type:
$$y' = p(x) \cdot y + q(x) \cdot y^{n}$$
Rearrange as follows:
$$\frac{y'}{y^{n}} = p(x) \cdot \frac{y}{y^{n}} + q(x) \cdot \frac{y^{n}}{y^{n}}$$
$$\frac{y'}{y^{n}} = p(x) \cdot \frac{1}{y^{n-1}} + q(x) $$
Now substitute $v=\frac{1}{y^{n-1}}$, and thus $v' = (1-n) \frac{y'}{y^{n}}$:
$$\frac{v'}{1-n} = p(x) \cdot v + q(x)$$
$$v' + (n-1)p(x) \cdot v = (1-n)q(x) $$
The last equation is a linear ODE in standard form.

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