What is the solution of the differential equation dy/dx = y/x?

₹0.0

Question: What is the solution of the differential equation dy/dx = y/x?

Options:

Correct Answer: y = Cx

Solution:

This is separable. Integrating gives y = Cx.

What is the solution of the differential equation dy/dx = y/x?

What is the solution of the differential equation dy/dx = y/x?

Step 1: Start with the differential equation dy/dx = y/x.
Step 2: Recognize that this equation is separable, meaning we can separate the variables y and x.
Step 3: Rewrite the equation as dy/y = dx/x. This separates the variables.
Step 4: Integrate both sides. The left side becomes ∫(1/y) dy and the right side becomes ∫(1/x) dx.
Step 5: The integral of 1/y is ln|y| and the integral of 1/x is ln|x|. So, we have ln|y| = ln|x| + C, where C is the constant of integration.
Step 6: Exponentiate both sides to eliminate the natural logarithm. This gives us |y| = e^(ln|x| + C).
Step 7: Simplify the right side. e^(ln|x| + C) can be rewritten as |x| * e^C. Let C' = e^C, so |y| = C'|x|.
Step 8: Remove the absolute value by considering the constant C' can be positive or negative. Thus, y = Cx, where C is a constant.

No concepts available.