Engineering Mathematics II (ECC3002)

Basic Solution Concepts, & Separable ODEs

The Solution Concepts (E.g. 1) A differential equation (DE) E.g.

can be solved (in order to find its solution) through integration

i.e.

c can be any values e.g. 0, -100, 99, 2.75 etc. Hence this solution is known as a family solution

The Solution Concepts (E.g. 2) A functional equation E.g.

has its own differential equation (DE)i.e.

A functional equation can be a solution to its differential equation (DE)

Functional equation

Differential Equation

This function is a solution to its differential equation!!!!

𝑦 ′=𝜕 𝑦𝜕𝑡 =0.2 (𝑐𝑒¿¿0.2 𝑡)=0.2 𝑦¿

The Solution Concepts (E.g. 3) The functional equation E.g.

has its own differential equation (DE)i.e.

A functional equation can be a solution to its differential equation (DE)

Functional equation

Differential Equation

This function is a solution to its differential equation!!!!The solution y has an arbitrary constant c – hence solution y is a general solution!!!

Initial Value Problem (E.g. 3) Based on example 3, the following ODE:

has a general solution of:

What is the solution, given an initial value y of 0.5 at t = 0

(i.e. y(0)=0.5)?

Thus, the solution at the initial point is The solution y does NOT contain an arbitrary constant c at the initial point– hence solution y is a particular solution at the initial point!!!

𝑦 ′=𝜕 𝑦𝜕𝑡 =0.2 𝑦

Case Study: Exponential Growth (E.g. 3)

Based on example 3, the following ODE:

has a general solution of:

The c value may vary, at initial point:

𝑦 ′=𝜕 𝑦𝜕𝑡 =0.2 𝑦

Case Study: Exponential Decay (E.g. 4)

Based on the negative version of example 3, the following ODE:

has a general solution of:

The c value may vary, at initial point:

𝑦 ′=𝜕 𝑦𝜕𝑡 =−0.2 𝑦

-Have a brEAk-Are u confused?

(a) NO (b) YES

If (b) then your foundation on Derivation & Integration is

Shaky!! DANGEROUS!!

Separable ODE Assuming that an ODE consists of 2 different variables

x and y, both variables can be separated as such x is on the right and y is on the left (vice versa).

By separating x and y, we can integrate both sides independently in order to derive the general solution:

Thus: Please remember to always introduce c immediately after integration

Separable ODE (E.g. 5) The following ODE:

Can be separated into:

The solution through integration:

produces…..

Thus:

Please remember to always introduce c immediately after integration – otherwise you will get different resultsy = tan (x) + c

Recall: Recall:

Separable ODE (E.g. 6) The following ODE:

Can be separated into: The solution through integration:

produces… Thus:

Recall:

Separable ODE (E.g. 6 cont.) For:

where:

Thus: Substitute back into the original equation: Hence…….

Recall:

Separable ODE (E.g. 7) Solved the following ODE as its initial value (i.e. derive the particular solution):

Through separation:

The solution through integration:

produces…. Thus: The initial value solution:

𝑦 ′=−2 𝑥𝑦 𝑎𝑡 𝑦 (0 )=1.8Recall:

Assume:

Through initial value substitution:=1.8

Selected Exercises from Text Book

Solve the ODE:

Verify that y is a solution of the ODE. Solve the IVP.

Selected Exercises from Text Book (cont.)

Find the general solution.

Find the general solution. Solve the IVP.

Thought for the day…

“Time and tide wait for no man””

--unknown