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Solve the differential equations.Some of the equations can be solved by the method of undetermined coefficients, but others cannot.$$y^{\prime \prime}-8 y^{\prime}=e^{8 x}$$

Calculus 3

Chapter 17

Second-Order Differential Equations

Section 2

Nonhomogeneous Linear Equations

Johns Hopkins University

Campbell University

Harvey Mudd College

Baylor University

Lectures

12:49

Solve the differential equ…

13:13

13:35

07:04

Solve the given differenti…

00:50

Find the general solution …

02:07

dear students here we have a differential equation. And this is a non homogeneous differential equation. And we have to find the general solution. And we know that for unknown homogeneous differential equation we have particular integral and complementary function in its general solution. So these students we will use we will use and determine the method of undetermined nine coefficients. So so did the studio students. This will be So we can write, we can write differential equation is D squared y minus eight Y. It is eight Dy DX students. It is eight. Dy. Do you want to contribute to the Power eight X. So he has students. So here's students. If you take if you take their students, Why? And this government. So this will give de into D minus eight. And here it will be Why is equal to E. To the power X. So nowadays students we have to find the complementary functions. So for complementary function we need auxiliary equation. For auxiliary equation, we have to put DS. M. So it will be M -8 and it it will be equal to zero. This is the district. This is the so this is this is the And auxiliary equation. So here from this we can find the roots. So M will be equal to zero. And the Syrians here, M -8 will be equal to zero. So this will be M. is equal to be And will be equal to their students. eight. So now dear students, we will find the complementary functions. So the complementary function which did not by Y. C. It will be equal to, Do we see one plus dissidents here? It will be C one, C two E. To the power IT X. So now the students we have to find the we have to find the particular integral. So these genes for particular integral. We have to consider a trial solution. So so dissidents for trial solution here in the right hand side we have what we have to keep our ethics and we know that for an exponential the trial solution is the trial solution will be equal to the students. Let's consider this. The incident. It will be why will be equal to let why will be equal to C. 3? And this will be c. three and it will be X. E. To the power should be X. I need the power ethics. So this will be the trial solution. This will be the trial solution for what it is for E to the power IT X. So distance. Why we have not taken only ethics about ethics because here in the in the particular in the complementary function we have need to depart latex and we have to change it by multiplying it with them by multiplying it with the least number of X. Least value of X. So it will be X. To the power eight X. So dear students now. So nowadays students we have to we have to here we know that this is a second differential equation. So second. So. So this will give the options. We have to find the why prime? So why Prime will be here? It will be C three. Do students need to be C3 into E. to the Power eight x. Bless it. You'll be it X. E. To the power it. Excuse me. So nowadays students, we have to find them double differentiation of this discourse. So this will give Additionally, this will give C3 in two Into the incident. This will give we know that for this. It will be 82. The power to tax and plus test. This will be here. It will be the students we have to find them. You have to use the product rule. So, it will be 64 64 Year. Students exceed the Power it takes. And Plus Ds tended to be it will be This needed to be 80 to the power so distant. This will be here. So, if you find it will be the distance. Let's find it again. So, this will give so D. S students eight into when we have to find the differential. So, it will be giv given is uh it eight x 8 x. and plus destined. It will be plus mm hmm. And this one should be e to the power eight x soon. No. So no decision's why prime can be written is My friend. Can be your near C3 C 38 due to the power politics. And plus it will be 64 C three Xd to the bar eight X Plus this year. And it will be eight. You can see C3, you can see c. x. So now the students, so now their students from here now we have to find here, we have to put the value of Y. Prime and Y double prime. Indeed, differential equation this form. So dear students, if you put it put it here we have white double prime minus eight Y. Prime is supposed to eat the power yet X. So what is why prime day students, Y double prime? Is this C three Or 8 C3? Eight C three to the students. It is it will be equal to the experience. It it C three U. To the power N. Dx. Bless 64 C three X. E. To the power index In plus c. three et tu. The power it. So now what is what is different? What is why prime? So Y. Prime here it will be minus eight. And Y. Prime is this unit is C. Three about eight X. And to be Plus eight c. 3 X. E. To the power it. So these students from this now this will be equal to the students. It is equals to what it is equal to E. To the power eight X. We have put the value right here. So nowadays students, so now their students this can be so it will be eight C three E. To the power eight X plus 64 C three E. To the X X eight X Plus C38. You didn't go for it X. So here it will be minus eight C three E. To the power eight XD restaurants. And here it will be minus 64 C three X. E. to the Power eight x equals two. I need to report it. So now from here the students this will be this will videos and this and this will be canceled out and this one and this one will be canceled out. So the remaining term will video students here. This is the remaining term well being. So this in the remaining time will be here it will be It will be 8 C3 U. To the power N X. Is equal to. Now eat the power eight X. So from here we can find C three which is equal to the students which is equals to one divided by it. Which is he goes to the students when divided by it. So now these students we have the trial solution, we have the trial solution for So these dreams notice that notice that the students, we have this trial solution which is right where it is, this is which is this one and this is for the distance make notice that this is for the particular integral for the pie. So DS regions by putting the values here. This will give, This will give Y. of B will be equal to distant it is c. three next to the power it takes. So why P will be equal to exceed the power eight X. And we have to put the value of C. Three so it will be eat. So now their students, we know that the general solution is equal to the cost Y. C. Which is the complementary function plus Y. P. A particular integral. And why will be equal to the president's wife sees what This is. c. one. Bless This decision it is C1 plus plus C. Two E. To the power eight X. And this will be there soon. It will be plus X. E. to the Power eight x divided by it. So now this unit we can write it is So we can write a distant c. one plus this will give you one plus. Yeah. If you take decisions, if you take here need to deploy our IT access a common. So it will give C two plus X divided by. So this is the required tradition. This is the required answer. Which way. What? So this will be the required answer. Which we what

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