Differential Calculus

We are given: Let’s isolate : This equation is not easily separable. Instead, observe that the right-hand side contains terms involving , and a good substitution might help.

Step 1: Try the substitution:

So the left-hand side becomes: Thus, the differential equation becomes:

Step 2: Multiply both sides by to simplify: Left side is the derivative of :

So now write:

Move all terms to one side: Factor :

Now group terms:

Now define:

Then consider:

Integrating:

So:

Answer:

Let the curve be , and suppose it passes through the point , with the tangent at this point having slope .

The general equation of the tangent line at point is:

Let’s find where this tangent cuts the x-axis and y-axis.

• x-intercept: set in the equation:
• y-intercept: set :

So the intercepts are:

Now consider the segment AB. The midpoint of this segment is:

We are given that the point of contact is the midpoint. So equating coordinates:

Now solve both equations. Start with the first:

Multiply both sides by 2:

Now use this in the second equation:

Substitute : So both conditions are satisfied.

So we conclude:

This is a separable differential equation:

The general solution is:

Use the point to find the constant:

Answer: