SPECIALIST MATHS Differential Equations Week 1

Differential Equations The solution to a differential equations is a function that obeys it. Types of equations we will study are of the form:

Obtaining Differential Equations To obtain a differential equation from a function, we must: differentiate the function, then manipulate the result to achieve the appropriate equation.

Example 1 (Ex 8B1) Show that is a solution of the differential equation

Solution 1 Show that is a solution of the differential equation

Example 2 (Ex 8B1) Show that is a solution of the differential equation

Solution 2 Show that is a solution of the differential equation Solution

Example 3 (Ex 8B1) Show that is a solution of the differential equation

Solution 3 Show that is a solution of the differential equation Solution Now

Example 4 (Ex 8B2) Given is the solution of the differential equation Find a, b, c and d given

Solution 4 Given is the solution of the differential equation Find a, b, c and d given Solution:

Solution 4 continued

Example 5 (Ex 8B2) Find a, b, c, and d if is the solution of and

Solution 5 Find a, b, c, and d if is the solution of and Solution:

Solution 5 continued

Solution 5 continued again

Slope Fields The differential equation gives a formula for the slope its solutions. For example the differential equation gives an equation to calculate the slopes of all points in the plane for functions whose derivatives are. That is it gives the slopes of all points of functions of the form

Slope Field for f ‘(x) = 2x x=0 x=1x=2 x=-1 x=-2 y x

Slope Field Generator y’ = 2x for y = x 2 + c y‘ = 3x 2 for y = x 3 + c y’ = 2x + 1 for y = x 2 + x + c y’ = x y’ = y y’ = x + y

Example 6 (Ex 8C1) Solve the following differential equation

Solution 6 Solve the following differential equation Solution:

Example 7 (Ex 8C1) Solve

Solution 7 Solve Solution

Solution 7 continued

Solution 7 continued again

Euler’s Method of Numerical Integration We find the solution of a differential equation by moving small increments along the slope field Start at (x o,y o ), then move up the slope field and at the same time going out horizontally h to get to the next point (x 1,y 1 ). The smaller the value of h the more accurate the solution.

Euler’s Method

Fundamental Theorem of Calculus Using Euler’s method if we make the size of h very small then the y value of the point we approach is given by:

Example 8 (Ex 8C2) Use Euler’s method with 3 steps to find y(0.6) for the differential equation with y(0)=2 Find y(6) using the Fundamental theorem

Solution 8 Use Euler’s method with 3 steps to find y(0.6) for the differential equation with y(0)=2 Find y(6) using the Fundamental theorem Solution:

Solution 8 continued

Solution 8 continued again

This week Exercise 8A1 Q2, 3 Exercise 8B1 Q 1 – 7 Exercise 8B2 Q 1 – 7 Exercise 8C1 Q 1 – 7 Exercise 8C2 Q 1, 2