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- W63639249 abstract "This chapter discusses exact first-order equations. Some first-order ordinary differential equations can be integrated directly. If the given ordinary differential equation has the form dy / dx = N ( x , y )/ M ( x , y ), and N ( x, y ) and M ( x, y ) are such that ∂ M /∂ x + ∂ N /∂ y = 0, then dy / dx = N ( x , y )/ M ( x , y ) is said to be an exact ordinary differential equation. Such an equation can be solved exactly, though the answer may be in terms of an integral. The (implicit) solution will be of the form ф ( x, y ) = C , where C is an arbitrary constant. Differentiating ф ( x, y ) = C with respect to x and rearranging terms gives dy / dx = − ф x /ф y Comparing dy / dx = − ф x /ф y to dy / dx = N ( x , y )/ M ( x , y ) gives ф x = −N, ф y = M and hence ∂ M /∂ x + ∂ N /∂ y = 0 is satisfied." @default.
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- W63639249 date "1992-01-01" @default.
- W63639249 modified "2023-09-27" @default.
- W63639249 title "Exact First Order Equations" @default.
- W63639249 doi "https://doi.org/10.1016/b978-0-12-784391-9.50063-4" @default.
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