Solve the differential equation in Exercises 9–22.
13. (...

Question

Solve the differential equation in Exercises 9–22.

13. (dy/dx) = √y cos²√y

Accepted Answer

Welcome back everyone. In this problem we want to solve the differential equation of D Y D X equals root y multiplied by a sin squared off root y, where y is greater than 0, x is not equal to 0, and we want to express our answer in the form y equals f x. Now, to solve this differential equation, let's first separate our variables, OK, so given the derivative of y with respect to x, D Y D x to be equal to root y sin squared off root y, OK, then this means that 1 divided by root y multiplied by sin squared of root y. Dy will be equal to DX. And now we can integrate both sides to help us solve for why. Now, to evaluate the integral on the left, let's use the substitution method, OK? So solving for the integral of 1 divided by root Y sin square root Y. DUI, OK, let's set. You to be equal to root y. Then that means the derivative of you with respect to Y. Would be equal to 1 divided by 2 root y, which means that 2 d u equals 1 divided by root ydyy. So we can rewrite our original integral, OK. In terms of you, yeah. As The integral of 2 divided by sine square u with respect to you. Now if we evaluate here, then we'll end up with 2 multiplied by the integral of cosex squared u with respect to you. Which is equal to, well, let me to save space, let me put it beside it, OK, which will be equal to -2. OK, so if we put this back into our original integral, then on our left hand side we'll have -2u. Being equal to x plus some constant c1 because the integral of d x is x and c1 is the constant of integration. Now remember u is equal to root y. So really this is negative2 multiplied by the cut of root y that's equal to x plus c1. I know if we divide both sides by -2, that means cut rooty will be equal to -C1 minus X, OK, divided by 2, and if we change the constant c1 into C. OK, then that means it equals C minus X divided by 2. If cut root y or cut root Y is equal to c minus x divided by 2, OK, then that means rooty would be equal to the inverse cotangent of C minus x divided by 2. I know if we square both sides, then y equals the inverse cotangent of C minus x divided by 2, all squared. Therefore, this is the solution to our differential equation. Thanks a lot for watching everyone. I hope this video helped.

Solve the differential equation in Exercises 9–22.13. (dy/dx) = √y cos²√y

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Solve the differential equation in Exercises 9–22.
13. (dy/dx) = √y cos²√y