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What is the domain of f(x) = 5^x?
2 answers:
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Answer:
Step-by-step explanation:
The original equation is . We propose that the solution of this equations is of the form
. Then, by replacing the derivatives we get the following
Since we want a non trival solution, it must happen that A is different from zero. Also, the exponential function is always positive, then it must happen that
Recall that the roots of a polynomial of the form are given by the formula
In our case a = 121, b = 110 and c = -24. Using the formula we get the solutions
So, in this case, the general solution is
a) In the first case, we are given that y(0) = 1 and y'(0) = 0. By differentiating the general solution and replacing t by 0 we get the equations
(or equivalently
By replacing the second equation in the first one, we get which implies that
.
So
b) By using y(0) =0 and y'(0)=1 we get the equations
(or equivalently
By solving this system, the solution is
Then
c)
The Wronskian of the solutions is calculated as the determinant of the following matrix
By plugging the values of and
We can check this by using Abel's theorem. Given a second degree differential equation of the form y''+p(x)y'+q(x)y the wronskian is given by
In this case, by dividing the equation by 121 we get that p(x) = 10/11. So the wronskian is
Note that this function is always positive, and thus, never zero. So is a fundamental set of solutions.
Green Line:
This is a horizontal line at x = 0, so it would be:
x ? 0
Since the shaded area is going to the left of 0 (this means x is negative):
x ≤ 0
Blue Line:
The shaded area is below the line, so:
y ≤ mx + b
When x = 0, y is -5.
y ≤ mx - 5
From each point, you go up 3 units, and to the right 4 units, so the slope is .