Solve the equation dy/dx = y^2 - x.

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Question: Solve the equation dy/dx = y^2 - x.

Options:

Correct Answer: y = 1/(C - x)

Solution:

This is a separable equation. Separating variables and integrating gives y = 1/(C - x).

Solve the equation dy/dx = y^2 - x.

Solve the equation dy/dx = y^2 - x.

Step 1: Identify the equation dy/dx = y^2 - x. This is a differential equation.
Step 2: Recognize that this is a separable equation, meaning we can separate the variables y and x.
Step 3: Rearrange the equation to isolate y on one side and x on the other. We can write it as dy/(y^2 - x) = dx.
Step 4: Integrate both sides. The left side will involve integrating with respect to y, and the right side with respect to x.
Step 5: After integrating, you will get an expression involving y and x. This will lead to a form that can be solved for y.
Step 6: Solve for y to get the final solution in the form y = 1/(C - x), where C is a constant.

Separable Differential Equations – The equation can be rewritten to separate the variables y and x for integration.
Integration Techniques – Understanding how to integrate functions and apply constants of integration.
General Solution of Differential Equations – Finding the general solution involves including an arbitrary constant after integration.