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Solve the given differential equation by using an appropriate substitution. The DE is a Bernoulli equation. Dy/dx=y(xy^6-1)

User Ron
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Final answer:

To solve the given Bernoulli equation, we can use the substitution v = y^(1-n), where n is the power of y. After applying the substitution and using separation of variables, we can obtain the solution. However, the solution process may involve complex calculations.

Step-by-step explanation:

To solve a Bernoulli equation, we can use the substitution v = y^(1-n). In this case, n = 6 (from the term y^6). Applying the substitution, we get the new equation dv/dx = (xy^6 - 1)v. This is now a linear first-order differential equation which can be solved using standard techniques such as separation of variables.

Next, we can separate the variables by writing the equation as dv/v = (xy^6 - 1)dx. Integrating both sides, we obtain ln|v| = (x^2y^6/2 - x) + C, where C is the constant of integration.

Finally, we can solve for y by substituting back the original variable and solving the resulting equation. However, please note that the solution process may involve some complex calculations and therefore it's recommended to use a software or calculator to find the final solution.

User Tsurahman
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