Calculus

dy/dx = (x + y)/x, using the homogeneous substitution v = y/x, what is y'?

Step-by-step solution with explanation

Final Answer

After substitution, , and the full equation simplifies to

Step-by-step solution

Rewrite the equation clearly

We start with the given differential equation. Our goal is to use the substitution v = y/x to simplify it.

Split the right side into two fractions

We split the fraction so we can see the y/x term clearly. This shows the equation is homogeneous because the right side can be written purely in terms of v = y/x.

Apply the substitution v = y/x

We replace y/x with v directly in the equation. The right side becomes 1 + v, which is much simpler.

Express y in terms of v and differentiate

Since v = y/x, we can write y = vx. Using the product rule to differentiate y = vx with respect to x gives dy/dx = v + x(dv/dx).

Set the two expressions for y' equal

Both expressions equal dy/dx, so we set them equal to each other. This is the key step that lets us solve for dv/dx.

Simplify to isolate dv/dx

Subtract v from both sides and the v terms cancel perfectly. We are left with x(dv/dx) = 1, a very simple separable equation.

Understanding this problem

Learning Insight

A homogeneous ODE is one where dy/dx depends only on the ratio y/x. The substitution v = y/x always converts it into a separable equation in v and x. This works because the v terms cancel, reducing the complexity of the problem.

Quick Tip

Whenever you see dy/dx = f(y/x), immediately substitute v = y/x. The v terms on both sides will always cancel, leaving a clean separable equation every time.

Common Mistake

Students forget to apply the product rule when differentiating y = vx. They write dy/dx = x(dv/dx) and drop the v term, which gives the wrong equation after substitution.