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Solve the initial value problem yy' = sqrt(x² y²) for x.

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Final answer:

To solve the initial value problem yy' = √(x²y²) for x, first separate the variables. Then, integrate both sides and simplify the result. Rearrange the equation to solve for y and find y = kx, where k is a positive constant.

Step-by-step explanation:

To solve the initial value problem yy' = √(x²y²) for x, we can begin by separating the variables. Rearrange the equation to get y'/√(y²) = 1/√(x²). Integrating both sides gives us ∫y'/√(y²) dy = ∫1/√(x²) dx. Simplifying the integrals gives us ln|y| = ln|x| + C, where C is the constant of integration. Rearranging further, we get ln|y| - ln|x| = C, which can be written as ln|y/x| = C. Exponentiating both sides gives us |y/x| = e^C. Since e^C is a positive constant, we can rewrite the equation as y/x = k, where k is a positive constant. Finally, solving for y gives us y = kx, where k is a positive constant.

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