Solve the differential equation in Exercises 9–22.
17. (...

Question

Solve the differential equation in Exercises 9–22.

17. (dy/dx) = 2x(y - 1), y > 1

Accepted Answer

Welcome back everyone. Solve the differential equation dyy divided by dx equals 4 x multiplied by y minus 5, given that y is greater than 5. So we're going to solve this equation using separation of variables. Let's go ahead and rewrite the original form. We have the y divided by dx equals 4 x multiplied by y minus 5. So in order to separate the variables we can simply divide both sides by y minus 5 and multiply both sides by dx, which gives us d y divided by y minus 5 equals. 4 x multiplied by d x and now that we have successfully separated the variables we can integrate both sides. So on the left hand side we have the integral of d y divided by y minus 5. Its value is ln of the absolute value of y minus 5. And since y is greater than 5, we can drop the absolute value and simply writes right. On the right hand side we can factor out 4 and integrate x dx. The integral of x dx is x 2 divided by 2, and we're going to add a constant of integration c because we have two indefinite integrals. So that ln of y minus 5. is equal to 2x2 + C. And now, if we want to identify the solution in terms of Y, we're going to take E to the left-hand side and the right-hand side. So, on the left-hand side, we're going to rewrite. Ellen of Y minus 5 in terms of e to the power of ln of y minus 5, and we're going to apply the same rule on the right hand side e to the power of 2 x 2 plus c. Using the properties of logarithms, we get y minus 5 on the left hand side, and on the right hand side we can simply write e to the power of 2 x 2 multiplied by e to the power of c using the properties of exponents. In addition, if we want to get the constant c at the end, it is better to write c1 for now. So let's go ahead and write c1 instead of c. And now let's suppose that e to the power of c 1 is equal to c, right? We're going to introduce a new constant so that y minus 5 is equal to c multiplied by e to the power of 2 x 2, and finally we can solve for y by adding 5 to both sides so that y is equal to. C multiplied by e to the power of 2 x 2 + 5, and that's our final answer for this problem. Thank you for watching.

Solve the differential equation in Exercises 9–22.17. (dy/dx) = 2x(y - 1), y > 1

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Solve the differential equation in Exercises 9–22.
17. (dy/dx) = 2x(y - 1), y > 1