Separable Differential Equations

Let u = 1 - 10y.
Then du = -dy, so dy = -du.
Substitute these into the integral.

	put the x’s back in

• Separate the variables.

• Integrate.

	subtract from both sides

Let Then the equation becomes

• If you are asked to find a particular solution, the next step would be to use
the additional information to solve for C . Since we are finding the general
solution in this problem, our answer will include the variable C .

• Solve for y.

	apply the exponential function to both sides
	ex and ln(x ) are inverses of each other

This means that either or . (Right? If you knew that
lyl = 2, then you’d know that y is either 2 or -2.)

When you write the general solution for this, give both solutions. Say, “The general
solution is or .

• Separate the variables.
In order to get all the x’s to one side , factor out an x from the right hand
side.

Now we can separate the variables

• Integrate.

To calculate the integral on the left, substitute u = 3 - 5y. Then du = -5dy
and we have

	put the y’s back
	subtract from both sides

Let Then the equation becomes

• If you are asked to find a particular solution, the next step would be to use
the additional information to solve for C. Since we are finding the general
solution in this problem, our answer will include the variable C.

• Solve for y.

	apply the exponential function to both sides
	ex and ln(x ) are inverses of each other

This means that either or . So we get either

or

The general solution is either either

or

if y = 4 when x = 0.
• Separate the variables.

• Integrate. To calculate the integral on the left, substitute u = y - 3. Then
du = dy. To calculate the integral on the right, substitute v = 2x + 1. Then
dv = 2dy and we have

	put the x ’s and y’s back
	subtract from both sides

Let Then the equation becomes

• Solve for C.

Substitute x = 0 and y = 4 and solve for C.

	the natural log of 1 is 0
C = 0

• Solve for y.

	use the law of logs that says
	apply the exponential function to both sides
	ex and ln(x ) are inverses of each other

This means that the general solution is either given by or
by