Warm Up 1.Find the particular solution to the initial value problem 2.Find the general solution to the differential equation

Sect. 6.3 Separation of Variables (Differential Equations) Use separation of variables to solve a simple differential equation.

Recall: Differential Equation A differential equation is an equation involving derivatives of an unknown function and possibly the function itself as well as the independent variable. Ex: y’ + 2y = 0 Ex: x 2 y’’ – 3xy’ + 3y = 0 The order of a differential equation is the highest order of the derivatives of the unknown function appearing in the equation 1 st order equations2 nd order equation

Differential Equations A function y = f(x) is called a solution to a differential equation if f(x) and its derivatives satisfy the differential equation. Ex 1: Determine whether the function is a solution to the differential equation y’’ – y = 0. a. y = sin(x) b. y = 4e -x c. y = Ce x, where C is any constant

a. y = sin x y’ = cos x and y” = -sin x y = sin x is not a solution. b. y = 4e -x y’ = -4e -x and y” = 4e -x y = 4e -x is a solution. c. y = Ce x y’ = Ce x and y” = Ce x y = Ce x is a solution. Verifying Solutions

You Try… 1. For the differential equation xy’ – 3y = 0, verify that y = Cx 3 is a solution. 2. Determine whether the function is a solution to the differential equation

Explicit Solution and Implicit Solution A function, defined explicitly as a function of an independent variable x is called an explicit solution. A function, defined in terms of both the independent and dependent variable is called an implicit solution of the equation.

Solving Differential Equations Example: In fact, there are many solutions to a D.E such as above. In the simplest cases, differential equations may be solved by direct integration to find a general solution.

To find a particular solution passing through a specific point in xy-plane, we need to impose a condition, known as: initial value, i.e. y(x 0 ) = y 0. This is known as the initial value problem. Ex 4: For the differential equation xy’ – 3y = 0, find the particular solution when x = -3 and y = 2. y = Cx 3 2 = C(-3) 3 The particular solution is Particular Solutions

Functions in Two Variables We will continue to look at first-order differential equations. We first looked at equations of the form y / = f(x). For example, expresses the derivative in the variable x alone. However, the equation is of the form y / = f(x, y), expressing the derivative in both x and y.

Separable Equations Given a differential equation if the function f(x,y) can be written as a product of two functions i.e. f(x, y) = g(x)  h(y), then the differential eq. is called separable.

Solving Separable Differential Equations 1.Rewrite the equation 2.Anti-differentiate both sides. 3.Plug in the initial condition and solve for C.

Example 5

Ex 6 Find a general solution to the differential equation Combined constants of integration

Ex 7 Find a general solution to the differential equation

You Try… Solve each differential equation:

Closure Explain the steps for solving a different equation that is separable.