0._Review_of_First_order_differential_eq

0._Review_of_First_order_differential_eq - Brushing-up Math...

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Brushing-up Math 251 for ChE 210 1. Separable equations Many first-order differential equations can be reduced to the form g(y) y’ = f(x) (1) By algebraic manipulations. Since y’ = dy / dx , we find it convenient to write g(y) dy = f(x) dx (2) This is merely another way of writing eq (1). Such an equation is called an equation with separable variables , or a separable equation , because in eq (2) the variables x and y are separated so that x appears only on the right and y appears only on the left. By integrating on both sides of eq (2), we obtain g(y)dy = f(x)dx + c (3) If we assume that f and g are continuous functions, the integrals in eq (3) will exist, and by evaluating these integrals we obtain the general solutions of eq (1). 2. Equations reducible to separable form 3. Exact differential equations 4. Integrating factors 5. Linear first-order differential equations A first-order differential equation is said to be linear if it can be written y’ + f(x) y = r(x) (4)
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