DETA - Separable Differential Equations Lesson

Separable Differential Equations

A separable equation is 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, dy/dx = g(x)f(y).

Separation of Variables

A technique for solving a differential equation that involves rewriting the equation so that each variable appears on only one side of the equation is called separation of variables. If dy/dx = g(x)f(y) and f(y) ≠ 0, then the equation written in differential form becomes dy/f(y) = g(x)dx. Let h(y) = 1/f(y) and the differential equation can be rewritten as h(y)dy = g(x)dx. Integrating both sides of the equation produces equation image indicator $LaTeX: \int h\left(y\right)dy=\int g\left(x\right)dx$ $\int h\left(y\right)dy=\int g\left(x\right)dx$ , from which a solution emerges. The presentation below illustrates the separation of variables technique yielding a general solution.

Finding a particular solution to a separable differential equation based on initial conditions is the theme of the next presentation.

Consider an alternative representation for a differential equation, M(x) + N(y) dy/dx = 0 where M is a continuous function of x and N is a continuous function of y. Just as with the previous notation, the x terms can be collected with dx, the y terms can be collected with dy, and a solution can be obtained by integration. The next presentation employs the separation of variables technique using the differential equation representation M(x) + N(y) dy/dx = 0.

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Homogeneous Differential Equations

Some differential equations that are not separable in x and y may be changed into separable differential equations by a change of variables. Differential equations of the form y'= f(x, y), where f is a homogeneous function are candidates for this technique. A homogeneous function of degree n is a function given by f(x, y) where f(tx, ty) = tⁿf(x, y) with n a real number. An example and a non-example of a homogeneous function of degree n are verified below.

description of homogenous and non-homogenous functions

A homogeneous differential equation is defined as an equation of the form M(x, y)dx + N(x, y)dy = 0 where M and N are homogenous functions of the same degree. View an example and a non-example of a homogeneous differential equation.

description of homogenous and non-homogenous differential equations

Once it is confirmed that a differential equation M(x, y)dx + N(x, y)dy = 0 is homogeneous, then it can be transformed into a differential equation whose variables are separable by the substitution y = vx, where v is a differentiable function of x. An example of solving a homogeneous differential equation follows: $Description of finding the generation solution of (x²-y²)dx-xydy=0$

The next two presentations provide additional illustration of how to solve first-order homogeneous differential equations.

Separable Differential Equations Practice

Separable Differential Equations: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of separable differential equations.

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