Solve the given differential equation by separation of variables. dy (2y+3)? dx 6x + 5...

Solve the given differential equation by separation of variables. $$\frac{d y}{d x}=\left(\frac{2 y+3}{4 x+5}\right)^{2}$$

Okay. We're solving a separable differential equation here. We're gonna get all the wires on the left hand side and all the X is on the right. Okay. There are no wise in this problem. So we're just going to kind of split the derivative. Think about multiplying both sides by DX. Okay, integrate both sides. When we do this. That is the rule. Okay. And the role Dy will be wine girl of signed five X. We know it is negative 1/5. Co sign five X. Okay. Plus constant plus C. And that is the solution to the differential equation.

In this question, we got to evaluate this differential uh Well we have to solve this differential equation by separation of variables. So let's do a cross multiplication here. Uh This this is a proportion so that everything but everything with the same variable comes over to one side. So we have five wineries. 24 D. Y. Is equal to dX. And this can be integrated right away because all the towns are not separated. Five is a constant comes outside integration of why it is to forest virus to 5/5. Integration of the existence. To express the concept of integral. Let's see. So that's five and five is cancer. Here we have virus to five and this is equal to X plus C. So this is the final answer. Thank you.

This question asked us to solve the differential equation three. X squared y squared Not what we know is that if this is r d y o ver de axe than r D y over, why squared is three x squared DX. This allows us to get the X is on the same side and the wise on the same side. No, let's take the integral. The integral of this is negative one over. Why be integral off this increased exports by one divide by the new exponents is execute. Don't forget our constant of integration plus C now lastly to right this just in terms of why we get wise negative one divided by X cubed plus sissy

Let's talk about this question. So you have to solve this differential equation by the separation of variables. So let's bring in all the variables over to one side. So the sign Y by cross multiplication can be brought, we're here so it's signed Y. Dy and dx is brought over to the other side with cross multiplication again. So that can be read it. And as one plus five X square dx. And or excess square remains after this. So this becomes sign Y. Dy remains as it is and this can be separated from the denominators common. So each the denominator should be given to each town with the numerator. So that becomes one of our extra square and this becomes five X square over x square. And uh this is the dx. Uh So this can be read it and assign widely Vira means that it is one over X squared and Britain, S x rays to minus two D X. And we're here. The sex square. Next square gets cancer fortunately. And now we can integrate both the sides. All the terms integration of sinus minus cause so we have minus cause y integration of x rays to minus one minus two is x rays to minus one over minus one. And we're here five is a constant. That comes outside and an integration of the excess just X plus the constant of integral. Asi so this can be now be read it and as minus costs Y is equal to x rays to minus one can be written as one over X. And this minus sign remains as it is. So 1/-1 is nothing but -1. Five X remains as it is plus the constant of integration. See? So this is the final answer. Thank you.