Question

1.Demonstrate with the aid of atleast 5 suitable examples the method of determining integrating factors of non-exact differential equations by visual inspection.

2. Describe using clear illustrations 5 typical applications of differential equations in engineering.

Answer 1

1)ans-Nonexact Linear Equation

The general form of differential equation is:

This equation may be reformulated as:

If this equation is exact, then the following equality holds:

The presence of this equality in the linear partial differential equation makes that equation exact.

The condition of exactness ensures the existence of a function F(x,y) such that:

Procedures

Let's take a moment to go over the procedure for solving these nonexact linear first order partial differential equations.

Step 1

Write the differential equation in standard form:

Step 2

Compute the integrating factor. The formula is:

The integrating factor is a function that is used to transform the differential equation into an equation that can be solved by applying the Fundamental Theorem of Calculus.

Step 3

Multiplying both sides of the equation from Step 1 by the integrating factor from Step 2:

Step 4

The left hand side of the equation in Step 3 is the derivative of:

This is after the chain rule for differentiation is applied. Recall that the chain rule for differentiation is the procedure for differentiating the composition of two continuous functions.

The equation from Step 3 may now be written as:

Step 5

Integrate both sides of the new differential equation from Step 4 with respect to x:

One may apply the Fundamental Theorem of Calculus to the left hand side of this equation to simplify it as:

lot of examples can be solved by above rules steps

2)ans--

Differential Equations are extremely helpful to solve complex mathematical problems in almost every domain of Engineering, Science and Mathematics.we have integrating and differentiating hundreds of equations throughout school and collage, because these equations have a hidden answer to a really complex problem. Mathematicians and Researchers like Laplace, Fourier, Hilbert etc., have developed such equations to make our life easier.Maxwell has given equations of Electromagnetic Field Theory, which help us to find the Magnetic Field or Electric Field easily by simple integration/differentiation. Various Transforms from Time Domain to Frequency Domain or vice versa in Engineering is only possible because of Differential Equations. In Mechanical/Civil Engineering, people use such equations for solving complex fluid dynamics problems, and finding the right balance of weights and measures to build stuff like a Cantilever Truss, for example. Partial Derivatives are used to find maxima and minima of functions with more than 2 dependent variables, while there are differential equations to find the complexity of a contour in case of Complex Numbers

lots of examples you can solve by above steps.