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Unformatted text preview: 9.3 Separable Equations 1. Separable Differentiable Equations Deﬁnition 1.1. A diﬀerentiable equation is an equation that involves an unknown function and one or more of its derivatives. Deﬁnition 1.2. The orderof a diﬀerentiable equation is the highest derivative in the equation Deﬁnition 1.3. A solution for the diﬀerentiable equation is a function that satisﬁes the diﬀerentiable equation Deﬁnition 1.4. A separable equation is a ﬁrst-order diﬀerentiable equation in which the expression for dy can be factored as a function of x times a function of y . dx In other words, it can be written in the form dy = g ( x) f ( y ) dx Example 1.1 (Example 1 on p.581). Solve
dy dx = x2 . y2 Example 1.2. Find the solution for the above diﬀerentiable equation that satisﬁes the initial condition y (0) = 2. √ dy x Example 1.3 (problem 2). Solve = y. dx e Example 1.4 (problem 12). Solve dy dx = y cos x , 1+y 2 y (0) = 1. 1 Section 9.3 Separable Equations 2 2. Orthogonal Trajectories Deﬁnition 2.1. An orthogonal trajectory of a family of curves is a curve that intersects each curve of the family orthogonally (at a right angle). Example 2.1 (Example 5 on page 583). Find the orthogonal trajectories of the family of curves x = ky 2, where k is an arbitrary constant. Example 2.2 (problem 30). Find the orthogonal trajectories of the family of curves y 2 = kx3 ...
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