Dear Team, I have understood the problem of solving first order differential equation by separation method as per study material. please check the exam , i stuck at the last few lines: we have y = e^(1/2)x^2+x+C =Ae^(1/2)x^2+x where A=e^C how come y(o)=2 =Ae^0 =2 A=2 SIMILARLY IN EXAMPLE OF INTEGRATING FACTOR METHOD: THE LAST 2 LINES as per study material: FInally , from the condition y(1)=0 we have: 0= 2-2+(C/e)= C = 0 so the required solution y(x) = 2- (2/x) please clarify regards Suresh sharma
Please clarify your references here, it will help us to see what's going on. Are you referring to a particular exam paper? - please say which one. And if it's in the study material, page number would be great! Which last two lines? (You say it's in Chapter 4, so that's good). Then we'll get back to you - thanks!
In both cases, it is the application of the initial condition that you are questioning. To solve a differential equation, we first come up with a general solution (the expression for y with the constant A in it in the first example, and the expression for y with the constant C in it in the second example), and then we use an initial/boundary condition to obtain the particular solution we're looking for. In these two examples, the initial/boundary condition is given in the question for you. In the first case, we are told y(0) = 2, ie y = 2 when x = 0. So we substitute x = 0 and y = 2 into our general solution for y, to work out the particular value of the constant A needed for our solution. Here, A = 2, and using this value of A gives our final answer. In the second case, we are told y(1) = 0, ie y = 0 when x = 1. So we substitute x = 1 and y = 0 into our general solution for y, to work out the particular value of the constant C needed for our solution. Here, C = 0, and using this value of C gives our final answer. When using these methods to solve differential equations to find expressions for probabilities in Markov jump processes later in the course, the initial conditions will not be given in the question. Instead, we will know the following: p_ij(0) = 0 if i is not equal to j (ie the probability of going from state i to state j in no time is 0). p_ii(0) = 1 (ie the probability of remaining in state i over a period of length 0 is 1 (the process doesn't move in no time))
