## MIT 1803 lecture 4

This lecture introduced two important forms that have known methods for solution.

The Bernoulli equation is $y' = p(x)y + q(x) y^n$ and can be solved by direct substitution by multiplying the equation through by $y^{-n}$, creating $v = y^{1-n}$ and eventually obtaining a linear ODE

$v' = (1-n) p(x) v + q(x)$

Homogeneous ODEs are in the form $y' = F(y/x)$ that is, the right-hand side is some function in the explicit form of y/x.  Note if the equation is scalable (“invariant under the zoom operation” as Mattuck puts it) in both variables (like checking for units in physics), if so, then it can be rearranged into this form.  Then let $z = y/x$, and of course then $y' = z'x + z$, substitute in, and the equation will become solvable by separation of variables.

Note that the physics model for large temperature differences is

${dT \over dt} = k (M^4 - T^4)$

Advice on approaching this:  substitute $T_1 = T/M$ which does several things.  $T_1$ and the equation is now dimensionless, $k_1$ is simplified as inverse time, and there is one less constant in the equation.