differential equation v/s algebraic equations

differential equation v/s algebraic equations   Okay, let's begin by considering a very basic differential equation : 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 the 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.   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…