MA 36600 LECTURE NOTES: WEDNESDAY, FEBRUARY 4 Exact Equations (cont’d) Exact Equations. Say that we have a frst order diFerential equation in the ±orm M ( x, y ) dx + N ( x, y ) dy =0 . Imagine ±or the moment that we could fnd a ±unction z = f ( x, y ) such that dz = ∂f ∂x dx + ∂f ∂y dy = M ( x, y ) dx + N ( x, y ) dy. Then dz =0i .e . , f ( x, y )= C must be a constant. This would be an implicit solution to the diFerential equation. The constantC can be ±ound by speci±ying that a point (x 0 ,y 0 ) be on the integral curve f ( x, y )= C . To recap, i± we can fnd a ±unction z = f ( x, y ) such that ∂f ∂x = M ( x, y ) and ∂f ∂y = N ( x, y ) then we say the frst order equation is an exact equation . This terminology is short-hand ±or saying “the diFerential equation is exactly the diFerential o± a ±unction.” Example. Consider the diFerential equation dy dx = − 2 x + y 2 2 xy . (This equation is neither linear nor separable.) Multiply both sides by N ( x, y )=2 xy to fnd the diFerential equation ° 2 x + y 2 ± dx +(2 xy )
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