Solving separable differential equations HW: Practice WS (last two pages of slide show)

Many differential equations are difficult or impossible to solve, and for those we create a slope field. But a subset of diff eq’s can be solved using simple integration techniques.

Example Since we don’t know how to solve this (yet), let’s create a slope field!

When y = 0, y’ = 0

When y = 0, y’ = 0

When x = -2 y’ = 0

When x = -2 y’ = 0

When x = -1 y’ = y

When x = -1 y’ = y

When x = -3, y’ = -y

When x = -3, y’ = -y

When x = 0, y’ = 2y

When x = 0, y’ = 2y

When x = 1, y’ = 3y

When x = 1, y’ = 3y

These slopes are getting mighty steep, and they are just going to get steeper! Let’s call it good for this slope field.

If we are given a point through which a particular solution passes, we could sketch that solution.

For example, the solution that passes through the point (0, 2) looks sort of like this

And the solution that passes through the point (0, -1) looks sort of like this

Notice that the line y = 0 (the x-axis) acts like a kind of horizontal asymptote, which no solution will cross. The family of solutions has a positive branch (above the x-axis), and a negative branch (below the x-axis).

Now let’s try solving this same diff eq explicitly, using the technique called “separating the variables”

Rewrite using Leibniz notation Separate the variables (and split the derivative into two differentials) so that y terms are on the same side as dy and x terms are on the same side as dx Integrate both sides Solving

Rewrite using Leibniz notation Separate the variables (and split the derivative into two differentials) so that y terms are on the same side as dy and x terms are on the same side as dx Integrate both sides To simplify things, let’s subtract the first constant from both sides and create a single new constant Remember, we are trying find a family of (continuous) functions, so we are solving for y. Remove the absolute value by writing the positive and negative cases separately Notice that the two cases mean that the family of solutions has two subfamilies; one positive and one negative. (This matches what our slope field indicated.) Exponentiate both sides to eliminate the natural log Also notice that it is considered extremely awkward to leave the constant of integration trapped inside the exponent. Luckily, it is easy to rewrite the solution so that the constant is in a more friendly location. (see next slide) Solving (notice two constants of integration)

We can write the two branches of the solution family more compactly by combing the positive and negative cases Use a rule of exponents to rewrite and begin removing the constant from the exponent Since C 3 is an arbitrary constant, that means that e to the C 3 Is also an arbitrary constant, so it is traditional to write the two branches of the solution family even more compactly by giving the leading constant a new name-- something like K Writing both branches of the family of solutions in simplified form

It would appear that K cannot equal zero, based on our solution process Can K = 0? But our slope field suggests that the function y = 0 is also a solution. Is it? So, y = 0 satisfies the differential equation (and matches our slope field), so our complete solution set is: If y = 0, then y’ = 0, and the equation becomes 0 = 0, which is true

Take the derivative of the solution and confirm that it satisfies the differential equation You already know how to confirm a solution to a differential equation, but let’s do it again. This matches our differential equation (substitute in y and you will see they are identical), so our family of solutions is valid. Confirming that Is a solution to Did you remember the chain rule?

we can find a particular solution if we are given an “initial value.” (a point through which the particular solution passes.) Once we have found the family of solutions This means that the particular solution we seek passes through the point (0, 2). Finding a particular solution to For example, if we are given that So the particular solution through (0, 2) is Graph this function on your calculator. We will compare to our slope field sketch. Notice that the initial value places this solution within the positive branch of the solution family. Now we can solve for K, using those x and y values.

On the TI-83On our slope field Not too shabby!

Now you can practice!

Answer key