Question

at2y’’ + bty’ + cy=0 (t>0,a,b,c are all real numbers)

equation obtained from substituting:

A) x=log(t)

B) t^2= e^x

Answer #1

We write the given differential equation by using substitution method .

Given the equation, a(second partial-∂ x/∂ t) + b(∂ x
/∂ t-first partial)+ c x = 0 show that A ⅇⅈ ω t is a solution for
certain values of ω (Be sure to specify all relevant values of ω).
(Begin by substituting x = A ⅇⅈ ω t into the above equation and
then solve for values of ω such that the equation is satisfied for
all values of A and t

Show that if a, b, c are real numbers such that b > (1/3)a^2
, then the cubic equation x^3 + ax^2 + bx + c = 0 has precisely one
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3. Consider a second order linear homogeneous equation Ay'' +
By' + Cy = 0 Suppose that e^at, e^bt and e^ct are solutions (where
a, b, c are constants). A. Show that e^at + e^bt + e^ct is also a
solution. B. Show that two of the numbers among a, b, c are
equal.

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solution to x′′−4x′−5x=0 by substituting it into the differential
equation. (Enter the terms in the order given. Enter c1 as c1 and
c2 as c2.)

Consider the BVP y′′=-cy, y(0)=0 where c>0, y(b)=0.
For each c, find b>0 such that the BVP has at least two
different solutions y1(t) and y2(t). Exhibit the solutions.

You know from class that, for any real numbers ?,?,?a,b,c such
that ?<?<?a<c<b and for any continuous function ?f, the
following equalities hold:
∫???(?)??+∫???(?)??=∫???(?)??,∫acf(x)dx+∫cbf(x)dx=∫abf(x)dx,
and
∫???(?)??=−∫???(?)??.∫baf(x)dx=−∫abf(x)dx.
Use these two facts to prove that, for any continuous function
?f,
∫???(?)??+∫???(?)??=∫???(?)??,∫stf(x)dx+∫trf(x)dx=∫srf(x)dx,
for all ?,?,?∈ℝs,t,r∈R (that is, in each of the following cases:
?<?<?s<r<t, or ?<?<?t<r<s, or
?<?<?t<s<r, or ?<?<?r<s<t, or
?<?<?r<t<s).

Consider the initial value problem, ay''+by'+cy=0, y(0)=d,
y'(0)=f where a,b,c,d and f are constants which one of the
following could be a solution to the initial value problem? Give
breif exlpanation to why the correct answer can be a solution, and
why the others can not possibly satisfy the equation.
a. sin(t)+e^t
b. cost+e^tsint
c. cost+1
d. e^tcost

Let
a, b and c be real numbers with a does not equal to 0 and
b^2<4ac. Show that the two roots ax^2+ bx + c =0 are complex
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15.)
a) Show that the real numbers between 0 and 1 have the same
cardinality as the real numbers between 0 and pi/2. (Hint: Find a
simple bijection from one set to the other.)
b) Show that the real numbers between 0 and pi/2 have the same
cardinality as all nonnegative real numbers. (Hint: What is a
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Let f(x) be a function that is continuous for all real numbers
and assume all the intercepts of f, f' , and f” are given below.
Use the information to a) summarize any and all asymptotes,
critical numbers, local mins/maxs, PIPs, and inflection points, b)
then graph y = f(x) labeling all the pertinent features from part
a. f(0) = 1, f(2) = 0, f(4) = 1 f ' (2) = 0, f' (x) < 0 on (−∞,
2), and...

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