Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Solve the given differential equations (a) by variation of parameters and (b) by the method of undetermined coefficients.$$\frac{d^{2} y}{d x^{2}}-9 \frac{d y}{d x}=9 e^{9 x}$$

Calculus 3

Chapter 17

Second-Order Differential Equations

Section 2

Nonhomogeneous Linear Equations

Missouri State University

Baylor University

Idaho State University

Boston College

Lectures

13:26

Solve each differential eq…

06:10

04:41

In Exercises 33-36, solve …

02:34

Solve the differential equ…

06:40

07:33

04:05

Solve the given nonhomogen…

the restaurants here we have a question and we have to solve this differential equation using variation of parameter method. So you notice that this is a non homogeneous differential equation. So it will have so it will have the particular integral Andy complimentary function in it. General solution. So D. S students picking right the So this is the working rule which you have mentioned here. So this is the differential equation this one. And and we have to find the auxiliary equation to find what the complementary function. And then we have to find the particular integral for that. We have to find your mm hmm. And for fixing geophysics. So the assyrians, we will use the Bureau's canyon of U. And V to find F. Of xn geophysics. So this will be the working rule. So let's do it first. First restaurant. We can write the differential equation in the form of it will be D squared y minus nine. Why is equal to nine x Divided by you to depart three expeditions. So now by taking what is the common? So this will give the Square -9. Why is equals to It will be nine x divided by U. to the power three expeditions. So from here. So from here this isn't this will give this will give uh we can write the auxiliary equation. So the auxiliary equation will be mm square minus nine is equal to zero. So these ones here we will find what we will find the we will find the roots. So the roots will be Here. It will be M. security calls to nine. So distance here the roots. So here the roots will be equal to M. Is equal to plus minus That's -3 d. instruments. So now we will find the complementary function. We should be C. One you to depart three X plus C two, E two depart minus three X C one H double three X plus C two, C two, E two minus three X. And and these students notice that here we have now to find what we have to find the particular integral decisions. So the particular integral is the formula is you which I've mentioned the working rule as well. So it will be U. F of X plus F of X plus V. G of X. So from here from here the students now we will find what what is you and what is V. So here notice that you is with use this one and V. Is this one because Because you hear I've mentioned in the working rule you is with C1 and vis with C2. So here you will be equal to the distance, you will be equal to E. To the power three X. And we will be equal to We will be equal to the distance to the Power -3 X. To know scenario students. We have to find what we have to find the of you envy. So the Canyon of Heroes Canyon can be written as you notice that the Rose Canyon zero skin and competition is U. V. In derivative of you derivative of V. This will be equal to district. This will be equal to putting the value of This one. This will give you three X -3 x. Find the derivative of three X. This will give three E. To the power of three X. Okay and here it will give minus three it is minus three U. To the power minus three executions. So from here so from here is since deputy will be equal to it will be equal to regions if you multiply them find the determinant. So this will become minus three. And here it will be one because anything power zero will be equal to zero because the basics are the same. So it will become into the power zero power zero means one so minus three into one. And and distance here it will be minus -3 to Devour one. Be simple. It. So here And the skin will be equal to -60 or so now. So now we have to find F. Of X. And G of X. So F of X. What is F of X? Distance F. Of X. Is you notice that F. Of X. Is minus integration of vi are it is V are divided by zero skin in the X. And and distance and G of X is equals to. So first we have to just find what if we fix so F of X will be equal to if if X will be equal to distance here. You know that what is v minus indication of ves VSE to diplomatic. three x. What is rds ST RST right inside of the differential equation which is here, this one which is nine X divided by two divided three X. It is Decent. It is. We can write it eight and 9 nine Youtuber -3 x divided by the Rose Canyon Canyon Is -6 divisions. So from here F of X will be equal to the distance f of X will be equal to minus integration. So it will be plus six So it will be three divided by two. Integration of integration of these students. It will be E. To the power here is also excited. I think I think here is X. Let's see the right inside Yes there is X. So it will be nine X -6. So from here this will be equal to uh this will be equal to X. E. To the power minus six expeditions divided uh and dx now we will use using the using the integration by parts we will find the integration which will be equal to which will be equal to the restaurants which will be equal to three divided by two in two. It will be three divided by two in two into minus X. U. To the power -6 x divided by six plus 1.6. Integration of U to the Power -6 X. The executions you have usd what we have used the integration by parts formula which is which is distance UV minus integration of V. D. You So here this U and V are not E. U. And V which you have which you have here. Yeah these are not these these these you envy distance. This is the general formula which is four integration by parts. So we will use here. So here you is what in our perspective here it is X. And the V. We have we have two integrated So we have to use the what? Integration by parts formula which is this one to find this one. So the assyrians. So so from here we can find F. Of X. S. F F X will be equal to the students. It will be equal to we can write it as we can write it as we can. If we simplify it this will be minus X. Jewelry minus X. E. To the power minus six X. Divided by four -1 of one 24 U. to the power -66 dear students this will be the so from here now so now here residents this will be equal to if you find distance. If you find him this can be furthermore Simplified in the form of it will be -6 X. E. To the power minus six x. And. And distance here it will be if you just take what we have to just take we have to just take what we have to just take the L. C. M. So this will be F. Of X. Now we have to find what we have to find G affects decisions. So for G of X. So G. D. S. Students C A G of X will be equal to integration of it. Is you? Sorry? Yeah it will be you are divided by, it will be you are divided by their Oskanian dx and we know that G of X will be equal to the students integration of you. You in our perspective to depart three x. And the students are which is six, it is nine x 9 x. To make sure of, it will be nine x. And you do the better -3 x Divided by the Rose Canyon which is -6 dx. So G. Of X will be equal to the students G. Of X will be equal to Ah this will be equal to -3 divided by two. Ah and M. D. S. Then this will be X. Dx. It will be integration of X. Dx distance. So now G F X will be equal to the Austrians, G F X will be equal to, it will be equal to minus three X squared divided by four. It will be it will be the experience, it will be equal to -3 X. Square Divided by four of the restaurants. So from here. So from here this weekend so now now we will find the we will find what you notice that now we have to find just make it here, we will write it here. So you know that way of be easy cause to here which I have already told you that here Y of B. We have find the value of U F F X V G of X. So now this will be here F of X. Is what is you the restaurants which is you UZ to depart three x into To be -6 Excuse to the bottom -6 x minus you develop minus six X Divided by it will be 24 and plus it is plus here I think yes You develop -3 x. And and the students what is F of X here? This is your fix G of axis, it is minus three X squared divided by for reasons. So from here why of P will be equal to So here from here, Y of P will be equal to -6 x. U to the power -3 x and minus E. To the power minus three X divided by 24 plus and it will be minus here, it will be minus three x squared to the power -3 x divided by for the issues. So from here. No so now these students from here we will find so these genes from here we will find we will find the general solution. The general solution is equal to Y. Is equal to Y of C. It will be equal to recent Y. Of C. Which is the complementary function plus the particular integral relation. So why will be equal to what is the particular integral? It is why is equal to E to the power It is, it is distance C one E. To the power three X plus C two. E two developed minus three X. And here it will be the complementary function which is Which is the students which is -6 minus six X. E. To the power minus three X minus E. To the power minus three X. Divided by 24 minus three X squared U. to the power -3 x divided by four. So this will be the general solution which we want. Or if you if you take uh if you if you want to take them Egypt government is X. Common is your desire. If you want to simplify it more. So this will be the general solution. Which way? What

View More Answers From This Book

Find Another Textbook

01:05

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and the…

01:13

08:29

Solve for the unknown, and then check your solution.$$\frac{3}{5} y-\fra…

02:24

Solve for the indicated variable.$$\text { For } x: 5 x-2 y=19$$

05:43

In each of the following exercises, solve the given inequality.$$6 x^{2}…

01:04

01:22

Complete the square in each of the following by putting in the form $(x+B)^{…

00:52

Find the unknown.$$r^{2}+21=5$$

00:47

Find the unknown.$$x^{2}+16=7$$

01:58

In each of the following exercises, solve the given inequality.$$(3 x+5)…