{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Separable

# Separable - Separable Equations We have found a strategy...

This preview shows pages 1–2. Sign up to view the full content.

Separable Equations We have found a strategy for solving first-order linear equations, and would now like to consider a particular class of first-order nonlinear equations. We will again make use of exact derivatives. Definition A differential equation that can be put in the form dy dx = g ( x ) f ( y ) is called separable . To solve separable equations, we can multiply by f ( y ) to get (1) f ( y ) dy dx = g ( x ) and observe that the LHS looks like the result of applying the chain rule to some function F ( y ) where F is an antiderivative for f . In particular, we have from the chain rule d dx F ( y ) = F 0 ( y ) dy dx = f ( y ) dy dx . So we can rewrite the LHS of (1) as an exact derivative to get d dx F ( y ) = g ( x ) and integrating both sides gives F ( y ) = Z g ( x ) dx. If G is an antiderivative for g then we have (2) F ( y ) = G ( x ) + C. Now from equation (2), we may or may not be able to solve for y as an explicit function of x . But even if we can’t, equation (2) still defines y as an implicit function of x , and is considered to be the general solution of the differential equation. In this case (2) is called the set of

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

Separable - Separable Equations We have found a strategy...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online