5

Solve the initial value problem: y' =csin €; y(0) = 1....

Question

Solve the initial value problem: y' =csin €; y(0) = 1.

Solve the initial value problem: y' =csin €; y(0) = 1.



Answers

Solve the initial-value problem. $$y^{\prime}+2 y=1, y(0)=1$$

But we have y prime plus two y equals one. Let's go ahead and figure out the integrating factor. So now we're gonna go ahead and multiply the differential equation through by E to the to T. And then we're going to rewrite the left hand side using product rule. So the left hand side is gonna look like derivative of each of the two t y. On the right hand side is each of the two t. Let's go ahead and integrating side. This gives us this. Now we're almost done. But we still need to solve for the value of C. So this is what why is equal to We just need to figure out the value, see, and we know that why had 001 It's also equal to one half plus c. So we concluded that C is equal to one half. That's it

My we want to be a separation of variables. The initial value problem. Why prime equals Y over X. Or differential equation Given the initial conditions, Why have one equals 12 So obvious separation of variables. We're gonna follow three steps where is gonna isolate our variables wine X on either side of the equation. So foot wide terms and left extras in the right. Then we're going to integrate both sides of the equation and solve such that we apply as function of X. This will leave us with some constant immigration. See So in step three we'll use our initial conditions to solve for C. R. A. And then we'll finalize our answer. So since why prime is simply dy dx we can isolate variables as Dy over Y equals D X over X, integrating both sides. Gifts L N Y equals X plus C. Constant immigration. If we expand into equal sides, we get Y equals X plus E or E T N X. Or a X. Next we saw percy. So one is equal to eight times one. Giving equals one giving a solution, Y equals X.

Hello. So today we're given in a question and were asked so for it. But also, this is an initial value problem. So you have to also use these given impact cram Attar's of X equals one and why equals one? You help Saul for the explicit value of this, um given problem. So we already have an inkling that this is going to be a separation separation of variables problem. The reason we know that is because we have a d. Y over the X, so that tells us that we have to separate the variables and to just X is wise or some other sort of variables. We do a substitution, and then we have to integrate that to figure out what the original equation is. Okay, so now that we know what we're doing, let's start working on this problem. Well, it's already writing in view every d x. So let's recall from the textbook that for these homogeneous type equations, uh, that first order systems we have, why equals the of X Times Act or the equals? Why react wire racks so we can take this knowledge of the substitution and we can plug it straight and to the given equation to try and simplify this expression. So we have the over the axe times the which is has ah, component of acts times acts equals two acts minus the axe all over X plus four times. No, The reason I right, the expression like that s right to be sub component x times access That just tells you that you have to use that, um product rule of differentiation, uh, to solve for this part. So what do I mean by that? Well, we have the original left side expression are vivax times the derivative of the right side expression one, my dad plus the, um, derivative of the left side expression times the original, the right side. So this tells us that the derivative of that is TV over DX. That's another way, Way too right. Be prime attacks times plus be and then that equals our right hand expression. Two x minus V X all over X plus four times x. Well, the first thing we see that we can do is track to be from the left side to the right side That gives us X TV over the X and then to, um, make it so that we can use this expression. Well, first, actually, let we see that we have a bunch of exes and all of these components, so let's pull in X out and simplify. So this is when you get to a certain part of the problem, you're like, Oh, wait, I should say something earlier. So we have X times two minus b all over X times one plus 40 to live X's cancel. That gives us two. Must be all over one plus for B so that I'm this expression simplifies down to that What we just saw for we have two must be all over one plus four B minus b. So now this makes it a little easier. So we have to multiply by one plus four B over one plus four B that the expression and then this right hand side then becomes too minus B minus B minus for B squared all over one for the well that can combine and become negative to be so. Then our right hand expression becomes, uh, negative for B squared minus two V minus two, all of her one plus four b. So now we have all the variables on the right side and we have some X variables on the left side so we can separate. This is when we separate our They're Ebel's, so we can multiply by the inverse of this expression. So 1/4 B all over negative for the squared minus to be nice to to both sides. And that gives us the one plus for B thank you for B squared minus to B minus two. TV equals, uh, one overact dx. Okay, well, now we can in Great. So this is the natural Lagerback's plus the natural legacy equals some complex expression on the left hand side. Well, when we have in a complex expression like this, that should make us think OK, well, have a b squared into the So let's try a u substitution. So you equals we'll set it equal the negative for B squared minus to B plus two to the denominator. We'll see if we can use this. So the derivative of this is negative eight, the minus two. And then that's all times DV Well, we don't have that on the top expression, but we see that we can actually fact arise they're factor. The top expression too negative too, can then be pulled out from both terms. And then that becomes, um, for V plus one. And uh huh, that's the expression we wanted. So we kind of use the Constitution. So we'll have to divide by negative to you to get our one for V to be. And then we can probable for these terms in for expression. And that gives us the integral of negative 1/2 times. Do you or one over you? Do you? Okay, well, that's a pretty straightforward expression. Toe integrates that negative 1/2 natural log of you equals the natural log of the X. See, Because when we have on the backs Plus, I want to see, we can combine this expression such that we have the natural log times, the inter components. All right, so we have this expression down here. Negative 1/2 you, Alan eu plus natural. Like a back seat. Well, what can we do? Well, one thing we can dio is raise both sides to the e power and, well, we have a constant from that becomes power. Uh, the inter variable. So we have you to the negative 1/2 equals accede or another way to write that is one over radical. You equals taxi. Okay, well, now, let's plug. Are you back in? So we have. Are you up here? One of these Red we have. Are you up here? And we're gonna plug it in to this guy. So we get one all over. We have a large Reichel. So we have needed for be square minus to B plus two. Double check that. Yes, that's correct. Okay. And that equals. See? Well, then we're down to our most reduced form of this expression. But we know from our original part of the problem that Y equals one. Max equals one. Well, we have some unknown sea component. Okay, well, we consult for that. What is me really equals. Why Over acts or one of her one. So V also, vehicles want so one all over. Negative. Four times one squared minus two times one plus two equals one time seat. So see, equals one Oliver radical negative for All right. So we know that. So let's try and right the finalized expression of this. So we have, um, are x times one over radical, That radical negative four in that equals one. And I should probably make a break so that we can see what we're doing. We don't get confused. All right? One over. Radical negative for times. Why? Over x squared? I asked to times why over acts was to so this could be rewritten such that we have one over thank you for Why squared over ax squared minus two. Why? Over acts plus two. Well, an interesting thing. Weaken dio make this, um, expression a little more palatable for us is we can want supply by the radical of X squared over radical back squared. But why are we doing that? To get rid of this dominator? Because we don't want this, um, a divisible part inside our radical. It just doesn't look nice. So what is radical X squared? Well, it's just x. So we have divide by radical negative four. Why squared? The X square is canceled minus two x y plus two X squared equals, uh, we can say Yeah, well, just keep one over. Radical negative four. No, we can divide by which is the same thing since that probably doesn't look very nice. so that's multiplied by one over acts on each side. That makes a little clearer that we can cancel out those axes. So the final expression for all of this is one over radical negative. Four. Why squared minus two X Y plus two X squared equals one over radical negative, for That's our final expression and then I'll zoom out so that we can see everything that we did. Teoh get to this point, that's it.

Here we have the second order differential equations why prime prime Plus one quarter, Y Is equal to zero. And the initial conditions Y Pie is to go to one and why primate pie Is also equal to -1. So here we write the characteristic equation was given differential equation that will be arab squared plus one quarter is go to zero. And solving for our we have r squared is go to minus one quarter which gives us her is the root of which is plus or minus the root of minus one quarter which is equal to plus or minus here. The route of one quarter, that would be half. So we have half and the root of -1. We get an eye there and this will be our Uh huh. And we can see that we have complex roots here. Since we have complex roots. Our equation will take the form Yeah. Y. T will be equal to C one, E. To the parliament T. Of course new new T plus C two E. To the poor lambda T. Sign new T. And here are new our lambda is equal to zero and our mule is equal to half. So mhm. Our general solution here will be whitey is he go to see one and the you will be arrested about zero. So it goes get one from the of course half of T plus C two plus C two. Sign half of G. And this is our general solution. Now that we have any initial conditions Which means we can find C1 and C2 now substituting the we have a derivative of why they're So here we find the derivative of why, which will be here we have course half of T Differentiating costs. We get a negative sign. So we have -11 sign of 50. And differentiating what's in the brackets will get here. Half over there. So here we have differentiating signs, We get a course it is positive, so we have c. two and that's of course half of T. And for the shooting was here we get half over there. So here we have differentiated and substituting this initial condition We get -1 is equal to and because of Mhm. Mhm. He will sign off pie. The sine of pi gives us um half of pi sign of half of pie gives us a one there so we get minus half C one plus. And here the course of half of pie it gives us a zero. So we don't have that part. And here the miners council out and get our I will see one is as to and how full the second initial boundary. We substitute um into a general solution pie and one for why? So we have one is equal to here. We have a course of half of pie. We get a zero there. So the C one. We already have C one so C two and sign off half of T which is pie of pie. We get one day so our C two will be go 2 to 1. now substituting this into our general solution we have our C1 is going to C two years ago to one. Therefore our general solution will be Y T. Is equal to C. one. We used to Yeah. Two because of half of T. Mhm. Yeah. Plus our city is one so we have signed have of cheap and this is our solution to this in each other. Value problems. Yeah. Mhm.


Similar Solved Questions

1 answers
Rectanguln; Lylinaricul Jnd Spherial Loordinutes Wmciterilted tripliritegralcl{(7,7*) trle: clin +;2 atticuver tFlc: rr xic: TacduuurdedEduraini MArETechricul;r puronni:Exariccicieco Hirihe -rcGengioandialemerId} find the intezral cf f Cv evalvatirg ore ofthe friple integrals
Rectanguln; Lylinaricul Jnd Spherial Loordinutes Wmc iterilted tripl iritegralcl{(7,7*) trle: clin +;2 attic uver tFlc: rr xic: Tacduuurded Eduraini MArE Techricul;r puronni: Exariccicieco Hirihe -rc Gengioandialemer Id} find the intezral cf f Cv evalvatirg ore ofthe friple integrals...
5 answers
Question 2Evaluate the integralstanh zdzcosh==10where the circles are positively oriented and traversed once.
Question 2 Evaluate the integrals tanh zdz cosh ==10 where the circles are positively oriented and traversed once....
5 answers
Prdc nether ELTAendetemed Icon (aloming Omojps Hcttrnlc cham Ca ~ont Jiz (Ertu NO E In 67y Unustd #nsrut Eunit)mor Ekcly t0Lntxn nydrocinorcemoicqulorcoioound nomechumed4HeliechenmelcmJncandGOJtou J6UW compbund nameEhnaflcMnenlnKuk"Raedet
Prdc nether ELTAendetemed Icon (aloming Omojps Hcttrnlc cham Ca ~ont Jiz (Ertu NO E In 67y Unustd #nsrut Eunit) mor Ekcly t0 Lntxn nydrocin orce moicqulor coioound nome chumed 4Helie chenmelcm Jncand GOJtou J6 UW compbund name Ehnaflc Mnenln Kuk" Ra edet...
5 answers
Esiabliah tho tildrain [email protected] o31apln Ina Idansry atan Wth Ma bl09 ccatalrng510 noco cdmiplcsled axp'euor cecu" clcO) 8a0 ` Fucn Clcli qutanly cacuAegtctnnabath A VJos &70 0f Iho 4No cot caxingRan OnLad e908* ~Zen 20NE Evra Odd UozVpie Bxttetrean Idenlily-Birtary
Esiabliah tho tildrain Koriity. @lall 2t6n29 To o31apln Ina Idansry atan Wth Ma bl09 ccatalrng510 noco cdmiplcsled axp'euor cecu" clcO) 8a0 ` Fucn Clcli qutanly cacu Aegtctnna bath A VJos &70 0f Iho 4No cot caxing Ran On Lad e 908* ~Zen 20 NE Evra Odd Uoz Vpie Bxttetrean Idenlily- Bir...
5 answers
If you dilute $25.0 mathrm{~mL}$ of $1.50 mathrm{M} mathrm{HCl}$ to $500 . mathrm{mL}$, what is the molar concentration of the diluted HCl?
If you dilute $25.0 mathrm{~mL}$ of $1.50 mathrm{M} mathrm{HCl}$ to $500 . mathrm{mL}$, what is the molar concentration of the diluted HCl?...
5 answers
#13 An arc of length 100 m forms a central angle Q in a circle of radius 50 m. Find the measure of @ in Tradians, radians, and degrees. 4 pts_
#13 An arc of length 100 m forms a central angle Q in a circle of radius 50 m. Find the measure of @ in Tradians, radians, and degrees. 4 pts_...
5 answers
SCCl;[email protected])DH €
SCCl; HO MgBr Excess @P)DH €...
5 answers
(40 pts) For each needed reacnon, Stereochemlstry = drawthe organic product(s} Note major and minor products must be included for stereoselective reactions:CHaMgB 2)Hjo'H;otNHz cat H"NOzAICI1KMnO 2) HzoNOzBrz FeBraHNOz "FizSO
(40 pts) For each needed reacnon, Stereochemlstry = drawthe organic product(s} Note major and minor products must be included for stereoselective reactions: CHaMgB 2)Hjo' H;ot NHz cat H" NOz AICI 1KMnO 2) Hzo NOz Brz FeBra HNOz "FizSO...
2 answers
Question 2 Consider the following joint density f(c,y) =c+1.5y2 ,for 0 < x < 1 and 0 < y < 1. Calculate E(X? Y = 1).
Question 2 Consider the following joint density f(c,y) =c+1.5y2 , for 0 < x < 1 and 0 < y < 1. Calculate E(X? Y = 1)....
5 answers
3" _ y _ 6y = 9(o) = " (0) = 0
3" _ y _ 6y = 9(o) = " (0) = 0...
5 answers
Paa-ACDm Ih Jamatct 07 120 A&t Cdo'oi30fu .Jo4' 105k49ey 0677t4]otulWa49$[nPIrenetio
Paa-ACDm Ih Jamatct 07 120 A&t Cdo'oi 30fu . Jo4' 105k49ey 0677t4] otul Wa 49$[n PIrenetio...
5 answers
QUESTION 8For which value of a the equation [ $" dz dy dx =2 does hold?a.3b.4c. 1d.2
QUESTION 8 For which value of a the equation [ $" dz dy dx =2 does hold? a.3 b.4 c. 1 d.2...
5 answers
H awittonsoanl ( guys 1 3 H What is the Friction 250 S up a "7 into
H awittonsoanl ( guys 1 3 H What is the Friction 250 S up a "7 into...
5 answers
(10 points) Find the unit tangent unit normal, unit binormal and curvature to the function r(u) (gin(2t),e' cos(24), 0 + 2t + 1} when
(10 points) Find the unit tangent unit normal, unit binormal and curvature to the function r(u) (gin(2t),e' cos(24), 0 + 2t + 1} when...
5 answers
Md E qquatlon cr tnc bntterit t0 Bc grend of f at Mr Indirard velurf(r) - IVz - 6; I = 7Ay-7-?(-1)Ev-7= Z6x-7)cv-7=7(x-7)Dv-1 = Z6x-7)
Md E qquatlon cr tnc bntterit t0 Bc grend of f at Mr Indirard velur f(r) - IVz - 6; I = 7 Ay-7-?(-1) Ev-7= Z6x-7) cv-7=7(x-7) Dv-1 = Z6x-7)...

-- 0.022247--