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Find the solution t0 the following di fferential equation:0 < t < 2) 2 < t < 6 t >+y=x(t)y(0) = 0x(t) =...

Question

Find the solution t0 the following di fferential equation:0 < t < 2) 2 < t < 6 t >+y=x(t)y(0) = 0x(t) =

Find the solution t0 the following di fferential equation: 0 < t < 2) 2 < t < 6 t > +y=x(t) y(0) = 0 x(t) =



Answers

Solve the given equation using an integrating factor. Take $t>0$. $$6 y^{\prime}+t y=t$$

Here we see that we have e to the natural log of two T as art integrating factor. So that's going to give us e to the T squared. Now let's go ahead and multiply this thing through the differential equation and we get e to the T squared. Why Prime plus two t he to the t squared. Why equals zero? And now we can rewrite this as the derivative with respect to t of e to the T squared. Why equals zero and we get you to the T squared. Why equal c? So that tells us that why equals c times e to the negative t squared? That's

Right. So let's certify substituting out these D squared R. D. T. S squared and the D. R. D. T. And the R turns out for using the substitution that I have on the right hand side of my screen. So you end up with R squared plus six are plus 34 equals to zero. And since I don't see any obvious factory, I'm gonna try the quadratic formula here. So I'll have negative B plus or minus the square root be square 36 minus four times A. Just one time. See 34 all divided by two times A. So that simplifies down to negative six plus or minus the square root of four times 34 is 1 36. So you're gonna have a negative 100 here. All about about two. And then we can simplify that down even further to be negative six plus or minus the square of 10. I divided by two. And so are our term is going to be negative three plus or minus five I. So these are complex solutions right here. So we're going to have a solution of the form why equals C. One E. To the negative three X. Co sign five X. Plus C. Two E. To the negative three X. Sign five X. And of course you can factor out the E. To the negative three X. And you end up with see one co sign five X plus. See to sign five X. So this would be your answer.

This question asked us to find the solution of the differential equation that satisfies the given additional condition. We know we have d'you over G is two teeth plus sequence where team divide by to you. Now let's get all the you terms on left inside to you D you and all the ti terms on the right hand side because it will make it significantly easier to integrate. Take the integral of both sides. Tiu to you becomes you squared because we increase the exploited by one and we divide by the new exponents. Marco, fishing just becomes one or two over too. On the right hand side, we have cheese squared plus tan of team plus c Remember, the integral seeking scored of tea is 10 of tea. Now that we have this, we know we're gonna be substituting. And if you have zero is negative five than t zero and use negative five substitute in. Remember, we're solving for C. We got C is 25. Lastly, plug back in to our equation that we determined once we integrated the sea is 25 instead of plus C, we now have plus 25 then take the squirt of both sides. Because we want this in terms of singular. You not. You squared and we end up with our solution. So this is all under the square root now?

This equation can be rewritten us or they are last T to t lost R squared plus t squared, Time TT Echo zero. No doubt for this part, it can be rewritten I asked half D r squared plus t squared, and now we can divide both sides by R squared plus t squared, and we can integrate both sides. Now they serve find no answer of this exercise.


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