Usage 1: Differential Equations - Example 2

FIRST A PRAYER: Hail Mary full of grace the Lord is with thee. Blessed are thou among women and blessed is the fruit of thy womb Jesus! Holy, Mary Mother of God pray for us sinners now and at the hour of our death! Amen!

For the differential equations of the type discussed below I will also provide a link to the other problems from fields like biology and engineering where the solution methods are relavent.

Hi, It's time to solve our second example problem. In it we shall verify that xsiny = cos y is a solution of (dy/dx)[xcoty + 1] = -1

As can be seen below, our solution is approached by diffentiation. The formula in question is differentiated with respect to x. Given that y is a function of x all y derivatives will have dy/dx along with them.

I shall discuss the methodology line by line. The first in is the differentiation step. Notice that it is done on both sides.

On the second line the result of the differentiation by the product rule is displayed. Note the dy/dx where a y derivative is taken.

In the third line I have differentiated the sin(y) term to get cos(y)dy/dx.

On the fourth line I multiply both sides by 1/sin(y) the result of which is ...

shown on the fifth line to establish the cos(y)/sin(y) term.

Then on the sixth line we use the identity cot(y) = cos(y)/sin(y)

Finally after collecting terms, we see our result (dy/dx)[xcoty + 1] = -1

We are going to have so much fun making math connections within and between disciplines! WE SHALL LOOK AT SEPARABLE EQUATIONS NEXT!!!

Now use the link to move onto Separable Equations!

Let's get to it ...D.E Ex3.

....or, the next link to get back to Differential Equations Part 1 for the next problem on the table.

Let's get to it ...D.E. Ex1.