Exact - Exact Equations In solving rst-order linear...

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Exact Equations In solving first-order linear equations and separable equations, we made use of “exact derivatives”. We now look at the most general setting in which we can use this idea to solve first-order ODE’s. We want to know when a first-order differential equation can be put in the form (1) d dx φ ( x, y ) = 0 , because from this point we can integrate both sides to get φ ( x, y ) = C. These are the solution curves, or if we can solve this equation for y ( x ) , this is the general solution. If a differential equation can be put in the form of equation (1) we call it an exact equation . I Example Any separable equation looks like f ( y ) dy dx = g ( x ) and can therefore be written in the form d dx F ( y ) - G ( x ) = 0 where F 0 = f and G 0 = g . So in this case we have φ ( x, y ) = F ( y ) - G ( x ) . Then the solution is F ( y ) - G ( x ) = C or equivalently F ( y ) = G ( x ) + C which agrees with our previous results. When is a given equation exact? To answer this question, we differentiate the left hand side of (1) using the multivariable chain rule to get d dx φ ( x, y ) = ∂φ
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