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We could leave the solution like this or we could write it in the form where K = 3C.However, then, we could combine these constants by writing C = C2 – C1.We could have used a constant C1 on the left side and another constant C2 on the right side.We write the equation in terms of differentials and integrate both sides:.Find the solution of this equation that satisfies the initial condition y(0) = 2.We use the Chain Rule to justify this procedure.In some cases, we may be able to solve for yin terms of x.
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Equation 2 defines y implicitly as a function of x.Then, we integrate both sides of the equation:.All y’s are on one side of the equation.To solve this equation, we rewrite it in the differential form.Equivalently, if f(y) ≠ 0, we could write where.The name separablecomes from the fact that the expression on the right side can be “separated” into a function of xand a function of y.In other words, it can be written in the form.A separable equationis a first-order differential equation in which the expression for dy/dx can be factored as a function of x times a function of y.Unfortunately, that is not always possible.ĭIFFERENTIAL EQUATIONS 9.3Separable Equations.It would be nice to have an explicit formula for a solution of a differential equation.We have looked at first-order differential equations from a geometric point of view (direction fields) and from a numerical point of view (Euler’s method).