2.6-2.7 - Sections 2.6-2.7 Exact equations Definition: M...

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Unformatted text preview: Sections 2.6-2.7 Exact equations Definition: M (x, y) + N (x, y)y = 0 is called an exact differential equation if there exists a function satisfying x (x, y) = M (x, y), y (x, y) = N (x, y), i.e, the solution of this equation is (x, y) = c. Notice that chain rule idea is embedded here! You don't need to pay too much attention to whether the given functions are continuous in a rectangular region or not. This is not the emphasis of this section. Follow these steps to approach an differential equation of the form M (x, y) + N (x, y)y = 0. Step 1: Calculate My and Nx to see whether they are equal. If not, this method is not applicable. If yes, go on to Step 2. Step 2: Depending on which one is easier, either integrate x M (x, y)dx or N (x, y)dy. I'll assume here we're doing the second one, as the first one is y done in your textbook. Step 3: (x, y) = y N (x, y)dy + h(x). Set x = M and find h(x) by integration. Step 4: Solution is given by (x, y) = c. Integrating factors This method is only applicable in practice when p = yN x is a function of x N -M only or q = xM y is a function of y only. Then you multiply the integrating factor to transform the equation into an exact form and follow the above steps. ( = exp pdx in the first case and = exp qdy in the second case.) M -N Euler's method There are two important equations you need to remember: *piecewise linear function approximation y = yn + f (tn , yn )(t - tn ) *Euler's formula yn+1 = yn + fn h You'll need to understand that accuracy generally improves as the step size h is reduced, but for a fixed step size h, the approximations do not necessarily 1 become more accurate as t increases. It depends on whether the solutions is a diverging family or not. Bonus! You can glance through the rest of the material in this section as long as you've mastered things mentioned above. 2 ...
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