Geometric Meaning of y’=(x,y) and direction fields Course Teacher: Vrajesh Sir Prepared by: Sanchit puranik Sarvaiya Pratik Shah Dharmil Mohmmad Ayaz Shivika Singh Birla Vishvakarma Mahavidhyalay Electronics and Telecommunication

What is differential equation? A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering,physics, economics, and biology.

Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Example: Multiply both sides by dx and divide both sides by y 2 to separate the variables. (Assume y 2 is never zero.)

A separable differential equation can be expressed as the product of a function of x and a function of y. Example: Combined constants of integration Separable Differential Equations

Family of solutions (general solution) of a differential equation Example The picture on the right shows some solutions to the above differential equation. The straight lines y = x and y = -x are special solutions. A unique solution curve goes through any point of the plane different from the origin. The special solutions y = x and y = -x go both through the origin.

Geometric Meaning of y’=(x,y) The ordinary differential equation of n th order can be written in general form as: F(x,y,dy/dx,d 2 y/dx 2,…..,d n y/dx n )=0. Let f(x,y,C1,….Cn)=0 be the general solution where C1,C2,…Cn are arbitrary constants. By giving particular values of these arbitrary constants, we obtain infinite number of particular solutions which represents an infinite number of curves, all of which satisfy the differential equation. Hence, an ordinary differential eq. n. represents family of curve.

Slope Fields Consider a differential equation dydx=f(x,y). Since the derivative is the slope of the tangent line, we interpret this equation geo metrically to mean that at any point (x,y)in the plane, the tangent line must have slope f(x,y). We illustrate this

with a slope field, a graph where we draw an arrow indicating the slope at a grid of points. The slope field for dydx=x+y+2 is illustrated at the right.

The solution to a differential equation is a curve that is tangent to the arrows of the slope field. Since differential equations are solved by integrating, we call such a curve an integral curve. This picture illustrates some of the integral curves for dydx=x+y+2.

You can see there are a lot of possible integral curves, infinitely many in fact. This corresponds to the fact that there are infinitely many solutions to a typical differential equation. The so

To specify a particular integral curve, you must specify a point on the curve. Once you specify one specific point, the rest of the curve is determined by following the arrows. This corresponds to finding a particular solution by specifying an initial value.

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