Worked examples and exercises are in the text STROUD PROGRAMME 24 FIRST-ORDER DIFFERENTIAL EQUATIONS

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Introduction Formation of differential equations Solution of differential equations Bernoulli’s equation

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Introduction Formation of differential equations Solution of differential equations Bernoulli’s equation

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Introduction A differential equation is a relationship between an independent variable x, a dependent variable y and one or more derivatives of y with respect to x. The order of a differential equation is given by the highest derivative involved.

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Introduction Formation of differential equations Solution of differential equations Bernoulli’s equation

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Formation of differential equations Differential equations may be formed from a consideration of the physical problems to which they refer. Mathematically, they can occur when arbitrary constants are eliminated from a given function. For example, let:

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Formation of differential equations Here the given function had two arbitrary constants: and the end result was a second order differential equation: In general an nth order differential equation will result from consideration of a function with n arbitrary constants.

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Introduction Formation of differential equations Solution of differential equations Bernoulli’s equation

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Solution of differential equations Introduction Direct integration Separating the variables Homogeneous equations – by substituting y = vx Linear equations – use of integrating factor

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Solution of differential equations Introduction Solving a differential equation is the reverse process to the one just considered. To solve a differential equation a function has to be found for which the equation holds true. The solution will contain a number of arbitrary constants – the number equalling the order of the differential equation. In this Programme, first-order differential equations are considered.

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Solution of differential equations Direct integration If the differential equation to be solved can be arranged in the form: the solution can be found by direct integration. That is:

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Solution of differential equations Direct integration For example: so that: This is the general solution (or primitive) of the differential equation. If a value of y is given for a specific value of x then a value for C can be found. This would then be a particular solution of the differential equation.

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Solution of differential equations Separating the variables If a differential equation is of the form: Then, after some manipulation, the solution can be found by direct integration.

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Solution of differential equations Separating the variables For example: so that: That is: Finally:

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Solution of differential equations Homogeneous equations – by substituting y = vx In a homogeneous differential equation the total degree in x and y for the terms involved is the same. For example, in the differential equation: the terms in x and y are both of degree 1. To solve this equation requires a change of variable using the equation:

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Solution of differential equations Homogeneous equations – by substituting y = vx To solve: let to yield: That is: which can now be solved using the separation of variables method.

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Solution of differential equations Linear equations – use of integrating factor Consider the equation: Multiply both sides by e 5x to give: then: That is:

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Solution of differential equations Linear equations – use of integrating factor The multiplicative factor e 5x that permits the equation to be solved is called the integrating factor and the method of solution applies to equations of the form: The solution is then given as:

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Introduction Formation of differential equations Solution of differential equations Bernoulli’s equation

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Bernoulli’s equation A Bernoulli equation is a differential equation of the form: This is solved by: (a)Divide both sides by y n to give: (b)Let z = y 1−n so that:

Worked examples and exercises are in the text STROUD Programme 24: First-order differential equations Bernoulli’s equation Substitution yields: then: becomes: Which can be solved using the integrating factor method.

Worked examples and exercises are in the text STROUD Learning outcomes Recognize the order of a differential equation Appreciate that a differential equation of order n can be derived from a function containing n arbitrary constants Solve certain first-order differential equations by direct integration Solve certain first-order differential equations by separating the variables Solve certain first-order homogeneous differential equations by an appropriate substitution Solve certain first-order differential equations by using an integrating factor Solve Bernoulli’s equation. Programme 24: First-order differential equations