# differential equation v/s algebraic equations

Okay, let's begin by considering a very basic

**: y prime equals cosine of x, and let's solve this equation. To solve means to find the unknown function y, and we can do this by integrating both sides of***differential equation***with respect to x. The integral of y prime dx equals the integral of the cosine of x dx, and that implies y is equal to the integral of cosine, which is the sine of x plus C.***the equation* Here, the constant of integration will be very important. So let's see if you can remember why we need it. For example, if C equals 1 then y equals sine of x plus 1, and the derivative of y is the cosine of x plus the derivative of one, which is zero because one is a constant. Therefore, y prime is just cosine of x. If C is a different number such as two thirds, then y equals sine of x plus two thirds, and again, the derivative y prime equals cosine of x. This is what we want: a function y satisfying y prime equals cosine of x, and these two functions are two different solutions. Since C can be any number, they are infinitely many solutions altogether.

What's different, then, about the solutions of

**and the solutions of algebraic equations. Let's consider the example from the last slide, y prime equals cosine of x, and an equation from algebra such as x squared plus five x plus six equals zero. With the differential equation, we saw that the solution was y equals sine of x plus C, and for each value of C, the solution is a function. On the other hand, if we solve this algebraic equation, which we can do by factoring, we get that x is either negative 2 or negative 3 which are numbers, as opposed to the solutions of the differential equation, which are functions.***differential equations* Now that we've solved a basic differential equation and have considered a key difference between differential equations and the more familiar algebraic equations, we'll set up and solve an important differential equation in the next video. See you then!