Determine the solution of the differential equation dy/dx = y^2 - 1.

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Question: Determine the solution of the differential equation dy/dx = y^2 - 1.

Options:

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

Solution:

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

Determine the solution of the differential equation dy/dx = y^2 - 1.

Determine the solution of the differential equation dy/dx = y^2 - 1.

Step 1: Start with the differential equation dy/dx = y^2 - 1.
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^2 - 1) = dx.
Step 4: Factor the left side: y^2 - 1 = (y - 1)(y + 1). So, we have dy/((y - 1)(y + 1)) = dx.
Step 5: Integrate both sides. The left side requires partial fraction decomposition.
Step 6: Write 1/((y - 1)(y + 1)) as A/(y - 1) + B/(y + 1) and solve for A and B.
Step 7: After finding A and B, integrate the left side to get ln|y - 1| - ln|y + 1| = x + C.
Step 8: Combine the logarithms: ln|(y - 1)/(y + 1)| = x + C.
Step 9: Exponentiate both sides to eliminate the logarithm: (y - 1)/(y + 1) = e^(x + C).
Step 10: Let K = e^C, then (y - 1)/(y + 1) = Ke^x.
Step 11: Solve for y: y - 1 = Ke^x(y + 1).
Step 12: Rearrange to isolate y: y(1 + Ke^x) = 1 + Ke^x.
Step 13: Finally, solve for y: y = (1 + Ke^x)/(1 + Ke^x).

Separable Differential Equations – The question tests the ability to recognize and solve separable differential equations by separating variables and integrating.
Integration Techniques – The solution requires knowledge of integration techniques to find the general solution after separation.
General Solution and Constants – Understanding the role of the constant of integration (C) in the general solution of differential equations.