Solving First-Order Differential Equations A first-order Diff. Eq. In x and y is separable if it can be written so that all the y-terms are on one side and all the x-terms are on the other

First-Order Differential Equations A differential equation has variables separable if it is in one of the following forms: Integrating both sides, the general solution will be : dy f(x) dx g(y) = OR g(y)dy - f(x)dx = 0

Separable Differential Equations Another type separable differential equation can be expressed as the product of a function of x and a function of y. Multiply both sides by dx and divide both sides by y 2 to separate the variables. (Assume y 2 is never zero.) Example 1

Separable Differential Equations Another type of separable differential equation can be expressed as the product of a function of x and a function of y. Example 1

Example 2 Separable differential equation Combined constants of integration

Example 2 We now have y as an implicit function of x. We can find y as an explicit function of x by taking the tangent of both sides. Notice that we can not factor out the constant C, because the distributive property does not work with tangent.

Example 3: Differential equation with initial condition – These are called Initial value problems Solve the differential equation dy/dx = -x/y given the initial condition y(0) = 2. Rewrite the equation as ydy = -xdx Integrate both sides & solve Since y(0) = 2, we get = C, and therefore x 2 + y 2 = 4 y 2 + x 2 = C where C = 2k

Example 4 Solve:

Solution to Example 4