# lec_3s - September 7, 2011 3-1 3. Separable differential...

This preview shows pages 1–3. Sign up to view the full content.

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.

Unformatted text preview: September 7, 2011 3-1 3. Separable differential Equations A differential equation of the form dy dx = f ( x,y ) is called separable if the function f ( x,y ) decomposes as a product f ( x,y ) = φ 1 ( x ) φ 2 ( y ) of two functions φ 1 and φ 2 . Proceding formally we can rewrite this as dy dx = φ 1 ( x ) φ 2 ( y ) dy φ 2 ( y ) = φ 1 ( x ) dx. Using the second formula, we can integrate both sides to get Z y dy φ 2 ( y ) = Z x φ 1 ( x ) dx + C as the general solution. Note that this is an implicit relation between y ( x ) and x . Indeed, the last integral formula has the form F ( y ( x )) = G ( x ) + C for some functions F and G . To find y ( x ) as a function of x we would have to solve this implicit relationship. This is frequently hard to do, so we will leave the solution in implicit form. A more general version of this is the d.e. M ( x ) dx + N ( y ) dy = 0 (1) We say that the general solution to this d.e. is an expression September 7, 2011 3-2 f ( x,y ) = C where f x = M (...
View Full Document

## This note was uploaded on 02/03/2012 for the course MTH 235 taught by Professor Staff during the Fall '11 term at Michigan State University.

### Page1 / 5

lec_3s - September 7, 2011 3-1 3. Separable differential...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online