Differential Equations

Today's question deals with differential equations. It's not very difficult, but it's very nice. Also, it provides a reminder for those who forgot how to solve those equations.

Solve: dy/dx = (2tan y) / x

In order to solve that and get a function, we first need to get each variable to where it belongs, and the appropriate "d notation" on its side. To do that, divide by tan y (to move it to the dy side) and multiply by dx (to move it to the x side):

cot y dy = 2/x dx

Now all the variables are sorted out, and we can easily integrate each side to get a function:
ln sin(y) = 2ln x + c (c is the arbitrary constant of integration)

This also equals:
ln sin(y) = ln x² + c

Since we want a function in the form of y = something x, let's make everything an exponent of e (to eliminate the ln's):
sin y = x² * e^c

Since c is an arbitrary constant, we can call it k and make it simpler:
sin y = kx²

Now, take the arcsin of both sides to eliminate the sine:
y = arcsin(kx²)

Have a great weekend,
Nadav

nadavs