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ssolve the given differential equation by using an appropriate substitution. the de is a bernoulli equation. dy dx = y(xy2 − 1)

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Final answer:

To solve the Bernoulli differential equation, transform it into a linear one through a variable substitution v = 1/y^2, leading to a linear differential equation that can be solved using the integrating factor method.

Step-by-step explanation:

To solve the given differential equation by using an appropriate substitution, observe that the equation has the form of a Bernoulli equation, which can be written as dy/dx = y(xy^2 - 1). We'll make a substitution to transform it into a linear differential equation. The substitutions commonly used involve setting v = y^-n, where n is the power of y in the non-linear term of y. Here, we take v = 1/y^2, which implies that dv/dx = -2y^-3(dy/dx).

Rewriting the equation in terms of v, we get dv/dx = -2(y^2(xy^2 - 1))/y^3. This simplifies to dv/dx = -2(x - y^-2). Now, we can substitute v back in to get dv/dx = -2xv + 2. The resulting equation is linear and can be solved using the integrating factor method.

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