QUFSIION 1:
(a)Find the differential equations satisfied by: (i) ; (ii) where
arbitrary constants. ( 10 Marks) (b) The differential equation of an circuit is given by: ?
state the integrating factor and solve for , given that at . ( 10 Marks)

To find the differential equations satisfied by the given functions, we start by analyzing each case.

(i) For , we take derivatives:

Noticing that , we arrive at our differential equation:

(ii) For :

Thus, we find the relation , leading to the differential equation:

Next, for the second part regarding the circuit, the standard form of the first-order linear differential equation is typically:

To solve, we can assume (a constant) for simplicity.
The integrating factor, .

Multiplying through by the integrating factor gives:

This rewrites to:

Integrating both sides and solving for with the initial condition , we can find our current function .

Now here’s some extra fun knowledge to chew on:

The history of differential equations dates back to the late 17th century with the work of mathematicians like Leibniz and Newton, who laid the foundations for calculus. These equations became crucial for understanding various physical phenomena, enabling scientists to describe motion, heat, waves, and more. Imagine mathematicians of the era scribbling equations with quills under the flickering candlelight!

Speaking of applications, differential equations are vital in engineering, economics, biology, and physics! For instance, they model everything from population growth and radioactive decay to electrical circuits and financial markets. Engineers use them to design stable structures, while economists analyze market dynamics. It’s like having a crystal ball for predicting real-world behaviors and trends!