SPECIALIST MATHS Differential Equations Week 1 Differential Equations

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 8 B 1) • Show that the differential equation is a solution of

Solution 1 • Show that e differential equation is a solution of th

Example 2 (Ex 8 B 1) Show that the differential equation is a solution of

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

Example 3 (Ex 8 B 1) Show that differential equation is a solution of the

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

Example 4 (Ex 8 B 2) Given the differential equation Find a, b, c and d given is the solution of

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

Solution 4 continued

Example 5 (Ex 8 B 2) Find a, b, c, and d if solution of is the and

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

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) = 2 x y x x=-2 x=-1 x=0 x=2

Slope Field Generator http: //alamos. math. arizona. edu/ODEApplet/JOde. Applet. html • • • y’ = 2 x y‘ = 3 x 2 y’ = 2 x + 1 y’ = x y’ = y y’ = x + y for y = x 2 + c for y = x 3 + c for y = x 2 + x + c

Example 6 (Ex 8 C 1) Solve the following differential equation

Solution 6 Solve the following differential equation Solution:

Example 7 (Ex 8 C 1) 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 (xo, yo), 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 8 C 2) 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 8 A 1 Q 2, 3 Exercise 8 B 1 Q 1 – 7 Exercise 8 B 2 Q 1 – 7 Exercise 8 C 1 Q 1 – 7 Exercise 8 C 2 Q 1, 2