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**Final Exam / Re: FE-3**

« **on:**April 17, 2013, 04:31:38 PM »

Also, I MIGHT have used 'x' instead of 't' on the test, would I lose marks if I used the wrong variable?

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Also, I MIGHT have used 'x' instead of 't' on the test, would I lose marks if I used the wrong variable?

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solve homogeneous-> roots are i,-i,2,-2

Yh=c1 e^2t +c2 e^-2t +c3 cost +c4 sint

Yp-> undetermined coefficients-> Yp= Atcost +Btsint +Ct +D

evaulate polynomial terms => C=-2. D=0

evaulate sinusoidal terms => A=1/10 B=0

Y= c1 e^2t +c2 e^-2t +c3 cost +c4 sint + (1/10)tcost -2t

Yh=c1 e^2t +c2 e^-2t +c3 cost +c4 sint

Yp-> undetermined coefficients-> Yp= Atcost +Btsint +Ct +D

evaulate polynomial terms => C=-2. D=0

evaulate sinusoidal terms => A=1/10 B=0

Y= c1 e^2t +c2 e^-2t +c3 cost +c4 sint + (1/10)tcost -2t

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find integrating factor: u=exp(integral tanxdx)=sec x

secx y= integral (tan x)= - ln (cosx) +C

y=-cosx(ln cosx) + C cosx

y(0)=1 => C=1

y=-cosx(ln cosx) + cosx

secx y= integral (tan x)= - ln (cosx) +C

y=-cosx(ln cosx) + C cosx

y(0)=1 => C=1

y=-cosx(ln cosx) + cosx

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Consider the second order equation

\begin{equation*}

x''=x^4-5x^2+4

\end{equation*}

(a) Reduce to the first order system in variables $(x, y, t)$ with $y = x'$, i.e.

\begin{equation*}

\left\{ \begin{array}{ll}

x'=\ldots\\

y'=\ldots\\

\end{array}\right.

\end{equation*}

(b) Find solution in the form $H(x,y)=C$.

(c) Find critical points and linearize system in these points.

(d) Classify the linearizations at the critical points (i.e. specify whether they are nodes, saddles, etc., indicate stability and, if applicable, orientation) and sketch their phase portraits.

(e) Sketch the phase portraits of the nonlinear system near each of the critical points.

(f) Sketch the solutions on $(x,y)$ plane.

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Are we expected to know how to use a Laplace transform to solve a non-homogeneous system?

This material is covered in chapter 6, which I do not know if we will cover by the end of the year. I cannot think of a type of system where a solution can only be obtained from this method, so I'm hesitant to learn it if we will always be allowed to pick which method to use when solving a non-homogeneous system.

More generally my question is, even if we do not cover it in class, how marginal will the value of this method be compared to undetermined coefficients or variation of parameters, on the final or future courses?

This material is covered in chapter 6, which I do not know if we will cover by the end of the year. I cannot think of a type of system where a solution can only be obtained from this method, so I'm hesitant to learn it if we will always be allowed to pick which method to use when solving a non-homogeneous system.

More generally my question is, even if we do not cover it in class, how marginal will the value of this method be compared to undetermined coefficients or variation of parameters, on the final or future courses?

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By inspection, $y = 1/2$ is a solution.

Darn, why did I not see this...

Just goes to show that slowing down during a test and looking at the question with a calm mind can do wonders

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Here is my solution:

where is says c1 in the final answer, replace that with c0 haha

where is says c1 in the final answer, replace that with c0 haha

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let y=e^rx

=> (r^4) +8(r^2) =16=0

(r^2 +4)^2 =0

roots= 2i, 2i, -2i, -2i

for double roots y2 and y4: y2=xy1and y4=xy3

y=c1 cos(2x) + c2 sin(2x) +c3 xcos(2x) + c4 xsin(2x)

solve I.C: y(o)=1 => c1=1

y'(0)=y''(0)=y(0)'''=0 => c2=c3=0 , c4=1

y=cos(2x) + xsin(2x)

=> (r^4) +8(r^2) =16=0

(r^2 +4)^2 =0

roots= 2i, 2i, -2i, -2i

for double roots y2 and y4: y2=xy1and y4=xy3

y=c1 cos(2x) + c2 sin(2x) +c3 xcos(2x) + c4 xsin(2x)

solve I.C: y(o)=1 => c1=1

y'(0)=y''(0)=y(0)'''=0 => c2=c3=0 , c4=1

y=cos(2x) + xsin(2x)

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