Qualitative Solutions To Differential Equations Qualitative Solutions to Differential Equations n differential equation we study the equations which are changed when their parameters changed or it is the study of anything that changes. From the study of differential, we learn that the derivative of a function nothing but is the rate of change of the function in calculus. So that any quantity likes velocity, temperature or volume that varies can be described by an equation involving its derivative. To study differential equations there are three main methods. The first method is analytic methods, in this method the solution of any differential equation is obtained by using a mathematical formula. The second one is numerical techniques, it provide an approximate solution for differential equation by using a computer or programmable calculator. Know More About Laplace Transform
And the third one is qualitative technique; in this technique we determining the general properties of solution without knowing the exact behavior. It is very difficult to find the analytical solutions to differential equations, often the behavior of the equations can be examined from qualitative types of behavior of quantity. Now we are going to the main topic that is Qualitative Solutions to Differential Equations. We investigate the behavior of the differential equation by the given expression: dM/dt = r M(K-M)/K, The above equation provides the basic information of a population where, K is the carrying capacity. Now, we rewrite the equation by introducing the new variable as follows, a=M/K, We use the method of separation of variables which helps to understand the solutions that is, dy/ dt = ra(1-a). This method is long and more complex, which involve integration by partial fractions and a lots of algebraic manipulation to reach the form of the solution and also difficult to understand the behavior of a (t) and also the whole population M (t). There are many cases in which no need to go through the elaborate the process if we want the appreciation solution of Qualitative behavior except the exact result or value. Learn More About Definition of laplace transform
Logistic Equation and generalization of the equation are found by using the above method, dM/dt = [rM(K-M)/K] â€“ H ,where H is a positive constant( H means harvesting). Lets we consider the case of no harvesting, i.e. set h=0. Then da/dt = ra(1-a) Where r is constant, which is a positive parameter and a is a variable that measures the carrying capacity of the population and this is a positive quantity. From above equation, here we can see the two solutions which correspond to da/dt=0. The values of a that satisfy ra(1-a) are a=0,a=1 . Because da/dt=0 so these values will not change. These values represent population levels that are constant in particular time. If we start the population at either of these values, it will stay at fixed level forever, according to our model. These values are called steady states and also sometimes known as fixed points or equilibrium. For other function when da/dt=0 then ra(1-a)>0 . The both r and a are positive, so that this whole term is positive if and only if 1-a>0. In other words a<1 then a, is starts in the range that is 0<a<1 will increase. Similarly we can calculate the range of a, values that result in decreasing a. Here we discuss on coupled systems on differential equations of the form:
n differential equation we study the equations which are changed when their parameters changed or it is the study of anything that changes....