Section 1.1: Integration Process

In our integration, C represents a constant of integration, as we are working with indefinite integrals. Additionally, we've performed a variable change to integrate the left side with respect to y, effectively eliminating the dx term. This simplification allows us to easily derive our solution—simply apply the inverse tangent.

Now we arrive at the general solution for our differential equation. To refine our solution, we can set C to 0 under specific initial conditions. For instance, if y equals 0 when x is 0, then C becomes 0, leading us to:

Section 1.2: Verifying the Solution

You might still question whether this solution fulfills the original differential equation, as I initially did. Thus, let's verify it. The derivative of the inverse tangent is crucial for this process.

By recalling the original differential equation, we can confirm that:

This holds true. A straightforward way to grasp it is to visualize a right triangle with side lengths and angles that correspond to our equation. Let's consider an arbitrary angle θ in such a triangle and build our understanding from there.

To find the cosine of θ, we need to square it. Assuming θ is the inverse tangent of x, we can express this as:

In a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. We'll select side lengths such that the ratio opposite/adjacent equals x.

To calculate the cosine of θ, we find the ratio of the adjacent side to the hypotenuse. The hypotenuse can be determined using the Pythagorean Theorem.

Next, we take the cosine of θ and then square it.

Remember that θ is the inverse tangent of x; hence, we make one final substitution:

This confirms our solution to the differential equation. Although this discussion has focused more on triangles than on differential equations, it's rewarding to see such a neat resolution to a non-linear differential equation.