Solve the differential equation_15+5u+3+ut...

Solve the differential equation.
$$
y=y^{\prime \prime}
$$

In this problem, we have to solve the differential equation. Why is it called toe? Why Double prime, which can be reading us. Why Double prime minus y is equal to zero. Now we can associate the characteristic equation Tau. This kind of differential equation, which is going to be T square minus one, is equal to zero. So the same place T plus one times T minus one is equal to zero, which in turn, in place either tease, tickle too negative. One Ortiz he called a positive one. So we see that the adults off the characteristic immigration are real and distinct. So whenever this happens, then I know that the general solution is going to be who icicle to see one times e to the politics plus C two times e to the power negative x because positive one and negative one are the roots of the characteristic incubation. And that's the solution

The first step to solving the differential equation is to multiply both sides by the integrating factor. So in this context, the integrating factor eed to the integral of negative one detox integrate negative one. We get e to the negative one X Now we know we're gonna multiply. We're gonna be multiplying this throughout the equation, Which means we have now each the negative acts. Do you? Why over jacks? Plus why times d of e to the negative axe divided by D backs is eat the X minus acts which is eating zero. Now we actually know anything to the earth. Power is one, regardless of what we have on the bottom sign, which means we now hop this and then remember, we could enter it with both sides with respect, tow X. So we simply have why I eat the night of maximal left inside is now the integral of one is X and I remember when we integrate, we always have the constant integration would just see. So this could be written simply in terms of why, as y equals acts plus c times E to the X

We need to find the general solution to the differential equation. Do you want to buy? DT is equal to eat to the teeth? So we start by separating the variables need to the T. D. Team and then we integrate both sides. We get why is equal to this integral is also E. T plus C. And they are asking for the general solution. So this is the general solution. The particular solution comes when you find see and that is if they give you initial conditions, so you can only find see if they give you some other amounts of information. The general solution is why is equal to E to the T plus C.

So for this problem, I started by multiplying both sides of my equation by D X. And so to complete our separation of variables, we can see that all we're going to have to do is divide both sides by each of the Y. So the left side of my equation I can write as e to the negative wipe our times, Do you? Why? Um and it's going to be equal to e to the x d x. And now that we have our separation of variables, we're going to be able to integrate the size of our equation. So when we do that are left side will become negative e to the negative. White power is equal to e to the X power plus C. And so our next step is going to be multiplying both sides of our equation by negative one. So when we do that, we get e to the negative wife is e to the X. Oh, excuse me. Negative e to the X, Um and we'll just continue to write plus C since just going to be a different constant there also, I'll change my first c two c zero so we can keep our second see, um, as just see. And finally, we are going to take the natural log of both sides of our equation to get a negative. Why is equal to the natural log of negative e to the X plus C. And finally, all we have to do is multiply both sides of our equation by negative one to get rid of this negative sign in front of the why and we get Our solution is why is equal to the negative natural log of negative E to the X power plus C.