5

Points) Find the solution of1Oy' + 2Sy = 24e"1with Y(O) = 8and Y (0) = 7....

Question

Points) Find the solution of1Oy' + 2Sy = 24e"1with Y(O) = 8and Y (0) = 7.

points) Find the solution of 1Oy' + 2Sy = 24e"1 with Y(O) = 8and Y (0) = 7.



Answers

Solve each equation.
$$
\sqrt{y}-7=0
$$

7/3 wide plus one equals zero. Subtract one, subtract one. So we get 7/3. Why equals negative? One multiplied by the reciprocal which is 3/7. So I have why equals negative three, seventh and

I want us. I want to solve this equation for Ah, Why here? So I want to get why on one side of the equation and I won't have everything else on the other. So because, um so let's take our why here that we have and let's try and move it to the other side of the equation, the right side. And I can do that by adding why I'm gonna undo the minus y with a plus y? And when I do that to the other side as well, whatever we do know one side of the equal sign, we have to do the other. So I have negative three acts so negative why plus positive. Why is just zero. So I won't write that plus seven equals and then zero. Plus, Bly is just Why So why equals negative three X plus seven

Yeah. In order to solve this equation, since it's already in factored form, we're going to take each factor and set it equal to zero so three y minus five equals zero and Y plus seven equals zero. Now we just need to solve each of these so we'll start by adding five and then divide by three. So our first solution is Y equals 5/3, and then our second solution, we get Y equals negative seven. And as always, these answers can be verified by plugging them back into the original equation for the Y values.

In this question, we are given the second order differential equation. Why prime prime plus seven Y. Prime Plus three Y is equal to zero. And to solve this first we write the characteristic equation. She's out squared plus seven hour Plus three is equal to zero. Now here we don't have or he has numbers that Give us a product of three in the summer seven. So we are going to use the quadratic formula to solve for the youth of our his mother wanted to is given by minus B plus or minus the root of B squared minus four A. C. All over to a. Now our A is equal to one, Our B is going to positive seven And our c. is equal to three substituting these values into our equation. We get -7 plus or minus the root of seven squared -4 x one x 3 all over to buy one And this gives us -7 plus or minus. So here we have 49 -12 which gives us 37. So we have the root of 37 here over to and simplifying this, we have minus 7/2 plus or minus 1/2. Apply by the root off 37. So here we have two distinct and real roots. These routes are you make no mistake here because many students would assume that since we have a roots and and it looks complicated that uh we have a complex route. So here this is just a real number and solving for this equation. We used the form for when the when we have free and distinct routes. So here we have for you and distinct the area and distinct. So we used the solution. Whitey will be c. one e. To the poor Erdogan T plus C. Two E. To the power to T. And putting our roots into this solution. We have C. one E. to the power. So here we have -7/2 plus Half of the Root of 37. And this is not applying with tea plus C two E. To the power -7/2 minus half Of the root of 37. And yeah, don't forget to put the tea. And this will be our general solution and final answer.


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