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Separable Differential Equations

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Separable differential equations are a class of first-order differential equations that can be analytically solved by separating variables and integrating. They are essential in modeling dynamic systems in science and engineering, such as chemical reactions in the atmosphere, drug distribution in pharmacokinetics, and investment growth in economics. The process includes identifying constant solutions and employing separation of variables for non-constant solutions, considering the domains of the functions involved.

Exploring the Basics of Separable Differential Equations

Separable differential equations are a subset of first-order differential equations that are amenable to analytical solutions, distinguishing them from other types that may necessitate numerical approaches. These equations can be reformulated to the form \(y' = f(x)g(y)\), allowing the variables \(x\) and \(y\) to be separated into functions exclusively dependent on each respective variable. This separation facilitates the integration of each side of the equation independently, leading to the general solution. The key to separability lies in the ability to isolate the terms involving \(x\) from those involving \(y\), thus dividing the equation into a product of two functions, each solely a function of one variable.
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Recognizing Separable Differential Equations

To ascertain if a first-order differential equation is separable, one must endeavor to express it in the separable form \(y' = f(x)g(y)\). Consider the equation \(y' = (x^2 + 9)5y\), which is separable as it can be rewritten with \(f(x) = x^2 + 9\) and \(g(y) = 5y\). Sometimes, an equation may not appear separable at first glance, but with algebraic manipulation or the use of logarithmic identities, its separable nature can be unveiled. For instance, the equation \(y' = xy + 3x - 2y - 6\) can be rearranged to \(y' = (x - 2)(y + 3)\), confirming it is separable. However, some equations, such as \(y' = \ln(x + y)\), are inherently non-separable due to the inseparable combination of variables within the logarithmic function.

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00

Form of separable differential equations

Expressed as y' = f(x)g(y), allowing separation of variables.

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Separation of variables in separable differential equations

Isolates x-dependent terms from y-dependent terms for independent integration.

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General solution of separable differential equations

Obtained by integrating separated functions of x and y independently.

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