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

Calculus 3

Chapter 17

Second-Order Differential Equations

Section 2

Nonhomogeneous Linear Equations

Campbell University

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

07:33

Solve the differential equ…

06:40

04:05

Solve the given nonhomogen…

04:13

07:19

Okay, so here we have a non homogeneous differential equation and we will solve this by the first matter that is, under the man coefficient matter. So first of all right, the auxiliary equation or the homogeneous system So we can right here are square minus R is equal to zero, therefore are multiplied by R minus one is zero. So our two routes are art is equal to zero on our physical do one now for two. Really roots. Our complimentary solution is given by sea when it is 20 x plus. See to it has to Thanks. Here it is 20 X statistical toe one. So here we can write C one plus c to it s two X. That is, to your compliment a solution. Right. But as we can see on the right side, off the d, we have areas two X right. And in our complimentary solution our solution is also either as two x right. Therefore, we have to multiply X right, So we can right here. Our particular solution is in the formal a zero multiply by x into areas two x right now take a derivative. So your whiteness is it going to zero mounted by my ex. It is two X plus. It has two X in here like a secondary ready that is zero multiplied by X ears to x plus years two x and again here plus us two X But here it is two x plus. It is to expect they will do to it is two x so we can right here too. It is two x directly right now substitute all these three values in this Cuban differential equation. Right, So you're yw slash That is a zero multiplied by X plus it s two x sorry x multiplied by India's two X And here y yes, that is a zero multiply by x. Here's to X plus two X And on the right side we have either two X right now Simplify this So we have a A zero x It is two x plus to a zero. It has two x minus Xerox. It has two X minus zero. It has two X now here these two times will begins allowed and here are two minus one that is equal to one. So we have only a zero Arrest two X is they called us to X right now compare the coefficient. Both sides. So here a zero is equal to one right now. Put the value off a zero in our particular solution. Right. So a particular solution is wip is equal toe one multiply by X. It is two X right? Therefore, general solution is given by why is it called toe y C plus y p Here y c is C one plus C to eat is two X and here Y p E E s E X s two x at least the Orson resolution right now solve this d using the radiation off parameter. So our particular solution was see one Sorry, why c is equal to C one plus C to eat us two X right? So as we can see here why one is equal to one And here why two is equal toe? It has two X right now take a deadly video. So your daily video off one is he called to zero. And here derivative off s two X studies. It is two X right now find their own skin W which is given by why one multiplied by y two days minus. Why? To multiply by. Why one does now here Why one is equal to one. Why two dances. It is two X hear why to is it is to X and y one. This is a call to zero. So if you're w is equal, do it is two X right? And on the right side of the we have a g affects the Col du Here's two X now our particular shoulders and is given by this formula. So here, why to that is it is two x y one that is one geo faxes. It s two x w is also it s too X. You're y one, that is one. Why do is us two x Geo faxes It is two x and W is also it has two X Now, here it is two X and it is to expel the cancel out. So here we have an integration off one only and he are one years to expel. Begins allowed. So here we have and only us to eggs. Now your integration off one that is on the X And here are integration off arrest to access. It is two X right, that is the our particular solution Now our general solution is given by y is equal to y c plus y p our complemented solution Waas C one plus C two it is two X and there were particular solution Is X it is two x minus. It is two X right. But here we can factor out here it is two x So your C one is entities here. We can factor out. It is two x so we can right here C to minus one. And this time is as it is now Here C to minus one So we can right here. See, two deaths is equal to C to minus one. Right? The we can take another constancy to this so we can right here. See to this multiplied by others to X plus X It is two X that is three orders under solution. Thank you.

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