Exact equations

Spotting an equation as the total derivative of a potential function, testing it with mixed partials, and reading solutions off level curves of that potential.

Notes on Diffy Qs: Differential Equations for Engineers, Exact equations

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3 exercises

Exercise 101

Solve the following exact equations, implicit general solutions will suffice:

$\cos(x)+ye^{xy} + xe^{xy} y' = 0$
$(2x+y)\, dx + (x-4y) \, dy = 0$
$e^x + e^y \frac{dy}{dx} = 0$
$(3x^2+3y)\,dx + (3y^2+3x)\, dy = 0$

2.5

Exercise 102

Find the integrating factor for the following equations making them into exact equations:

$\frac{1}{y}\, dx + 3y \, dy = 0$
$dx - e^{-x-y} \, dy = 0$
$\left( \frac{\cos(x)}{y^2} + \frac{1}{y} \right) \, dx + \frac{x}{y^2} \, dy = 0$
$\left( 2y + \frac{y^2}{x} \right) \, dx + ( 2y+x )\, dy = 0$

2.5

Exercise 103

Show that every separable equation

y' = f(x)g(y)

can be written as an exact equation, and verify that it is indeed exact.

Rewrite $y' = xy$ as an exact equation, solve it, and verify that the solution is the same as it was in an example in the source text.

2.5