1.In each problem below, an inhomogeneous system x%u2018 = Ax + b of two rst ord
ID: 3342623 • Letter: 1
Question
1.In each problem below, an inhomogeneous system x%u2018= Ax + b of two rst order ODEs is
given. The qualitative behavior of the solutions (as seen through their phase planes) near
the critical point are all dierent. The motivation of this problem is to discover how the
eigenvalues and eigenvectors determine the behavior of the solutions. For (a)-(e), carry
out the following steps:
(a) Find the equilibrium solution, or critical point, for the given system.
(b) Write the associated homogeneous equation, x%u2019 = Ax, and nd the eigenvalues and
eigenvectors of A.
(c) Draw a phase portrait centered at the critical point. (The PPLANE applet at
http://math.rice.edu/~dfield/dfpp.html is fantastic.)
(d) Describe how solutions of the system behave in the vicinity of the critical point (e.g.,
do they approach the critical point, depart from it, spiral around it, or something
else).
(a) x' = x- 4y- 4; y' = x - y- 6
(b) x' = 0.25x - 0.75y + 8; y' = 0.5x + y - 11.5
(c) x' =- 2x + y -11; y' = -5x + 4y - 35
(d) x' = x + y- 3; y' = -x + y + 1
(e) x' = -5x + 4y - 35; y' = -2x + y-11
2.Suppose that x1(t) = u(t) + iw(t) solves the system x'= Ax, where u and w are real-
valued vector functions. Show that u and w also solve x' = Ax. Explain your reasoning.
Explanation / Answer
ANSWER
THIS WILL HELP YOU
xy'-4y=x
y'-(4/x)y=x
m(x)=e^ %u222B -(4/x)dx
m(x)=e^-4ln(x)
m(x)=e^ln(x^-4)
m(x)=1/(x^4)
ym(x)= %u222Bm(x)x(dx)
y/(x^4)= %u222B(1/x^4)x(dx)
y/(x^4)= %u222B(1/x^3)(dx)
y/(x^4)= (x^-2)/(-2) +C
y/(x^4)= -1/(2x^2)+C
y= (-x^2)/2 +Cx^4
solve for C
6=(-(1)^2)/2 +C(1)^4
13/2=C
=> y=(-x^2)/2 +(13/2)x^4
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