![]() ![]() Solve the following initial-value problem: On the other hand, integral calculus connects the small pieces together and lets us know how much of something is made, on the whole, through a series of changes.\): Solving an Initial-value Problem Differential calculus breaks up things in several small parts and tells how they change from one instant to the next. Question 4: How many types of calculus are there?Īnswer: There are two kinds of calculus. Further, integral calculus deals with the inverse of the derivative, that is to say, to find a function when we know the rate of change. ![]() Question 3: Why do we study differential calculus?Īnswer: Differential calculus concerns with finding the instant rate at which one quantity changes with regards to the other, which we call the derivative of the first quantity with regards to the second. Moreover, separable equations have the form dy/dx = f(x) g (y) and are referred to as separable because one can bring the variables x and y to the opposite sides of the equation. Question 2: What are the differential equations?Īnswer: Separable differential equations are a very common type of differential calculus equation which is particularly quite simple to solve. Further, Calculus 3 covers multivariable calculus roughly. Similarly, Calculus 2 covers integral calculus approximately. Question 1: Is Calculus 1 differential calculus?Īnswer: Calculus 1 does cover differential calculus to an extent. This concludes our discussion on this topic of differential calculus. Let us now solve it for the genral solution – This differential equation is clearly a variables-separable equation. Where N – the number of radioactive atoms present in the sample at time t, k – proportionality constant (the radioactive decay constant). You must have read in Physics that the phenomenon of radioactive decay is governed by the differential equation – Here, we have replaced by the previous arbitrary constant c, by a new constant k, where k = -c. of the form, y = f(x) solution can then be given as – Where c is the constant of integration and must be introduced immediately after integration. Solution: Separating the variable, and integrating with respect to x – ![]() The solved examples below will introduce you to some more new concepts, and should be enough to set you on track! Solved Examples on Differential Calculus You must solve plenty of problems to make yourself proficient in applying it. That’s all you need to know about this method. This solution is only valid over the range of x, known as the domain of y(x). This is the solution of the differential equation in variables separable form. Now, y’dx is simply dy, so on the left-hand side, we can switch to y as the variable of integration. Integrate on both sides with respect to x Such an equation can be easily solved for its solution in the following way – The name must be clear from the format of the equation since all of the terms with the independent variable x are on one side, while all of the terms with the dependent variable y are on the other side. Where f and g are continuous functions This is known as the Variables Separable form of the differential equation. Many practically useful first-order differential equations can be reduced, by purely algebraic manipulations, to the form – Differential Calculus with Variables Separable ![]()
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