Question
5. Solve the given partial differential equation 3 marks
5. Solve the given partial differential equation 3 marks
Answers
Solve the given differential equations. $$5 y^{\prime \prime}-y^{\prime}=3 y$$
Yeah, we want to solve a given differential equation which is dy dx is equal to see can't why for the initial conditions Y of zero equals zero. This question is challenging our ability to solve an initial value problem in particular is challenging us to solve this IVP with separation of variables. There are four steps to execute first. We're gonna isolate are X and Y terms on either side of the equation. So why? In the last Excellent. Right. In step two we integrate both sides because each side has a differential dy dx step three we saw the integration techniques and in step four we saw the IVP plugging in our initial conditions. So in step one, isolating X and Y. I guess coast Y Dy equals the X. That's because Dy oversee can't Y equals coast Y. Dy that we have integral coach. Why do I equals integral? The X or solving sign Y equals X plus constant integration. See in step four we saw percy using our initial conditions, we plug in Y X equals zero. The sine of zero equals zero equals zero plus C, so C equals zero. And our final solution is sign of Y is equal to X.
Problem. Three of section 57.1 asked for us to solve the falling differential equation. So the first thing that we're going to do to solve this equation was we're going to separate the variables in this part. So to do so, we need to write why Prime Minister Y t X, and we're gonna set that equal to negative three. Why? So next what we're gonna do is we're going Teoh, multiply both sides by d. X, um, and then divide both sides by Why? So that leaves us with de y times. Why is equal to negative three d X Now we're going to integrate both sides, and doing so leaves us with Ellen of why the absolute value of why is equal to negative three X Now we need to get rid of the Ellen of why? And to do so, we put base E in the equation. 01 thing I forgot to mention. We need to add a plus c um, to this negative three x because obviously, when you enter, um, when you differentiate or you do the anti derivative, you need to add the plus c. Um, so after this stuff you should have a plus e And that leaves us with why is equal to e to the negative three x plus c? So we're gonna go appear and simplify that, um, riches. Why is equal to eat the negative three x times e to the C, and this could be replaced by new variable. We're just gonna call it a and that will leave us with wise equal to a times E to the negative three X Now, we're gonna go back to our initial condition, which says that at zero when X is equal to zero, why does he go to five? We're gonna substitute those values in insult for a So we get five is equal to eight times E to the negative three times zero. And as we know anything to the zeroth, power is equal to one. So we get a times one is equal to five, so a equals five. So the solution to this situation is Y is equal to five e to the negative three. Next
Problem. Three of section 7.1 asked for us to solve defying differential equation. So the first thing that we're going to do to solve this is we're going to rewrite wide prime, as do I d. X and doing so we're going to separate the variables. So we're going to multiply both sides by D X, and then we're going to divide both sides by why so that leaves us with de y over. Why is equal to negative three d X Now we're going to integrate both sides, leaving us with Ellen of the absolute value of y is equal to negative three x plus c. Now we're going to get rid of the Ln of why. And we need to put Basie on everything to do this. So and this this comes out to be why is equal to e to the negative three x times e to the C? No. The next thing that we can do is we can rewrite this as some very well A. So our equation now looks like this. Why is equal to a times E to the negative three X now what we're going to do is we're gonna go back to the initial condition and satisfy it. So we're setting. Why go to five and we're solving for a by setting X equal to zero and why go to five? So we get five is equal to eight times E to the negative three. I'm zero, as we know. Negative three times 00 And anything to the zeroth powers one. So that leaves us with a is equal to five. Meaning our final solution looks like this wise equal to five times e to the negative three x.
Problem. We're 18. We have five wise got a three y prime. Why primes Go to 5/3. Why were given? Why zeros you go seven. So therefore, P zeros. You go to seven K's goto 5/3. Then our function is wise to go to seven e to the five t over three.