Solving First-Order Differential Equations

1. 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

2. 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 =

3. Separable Differential Equations Another type separable differential equation can be expressed as the product of a function of x and a function of y. Example 1 Multiply both sides by dx and divide both sides by y2 to separate the variables. (Assume y2 is never zero.)

4. 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

5. Example 2 Separable differential equation Combined constants of integration

6. 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.

7. Example 3:Differential equation withinitial 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 4 + 0 = C, and therefore x2 + y2 = 4 y2 + x2 = C where C = 2k

8. Example 4 Solve:

9. Solution to Example 4