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nikdorinn [45]
3 years ago
9

Solve the decay equation y' = -ky using the integrating factors method. Show work

Mathematics
1 answer:
Alex777 [14]3 years ago
7 0

Answer:

y=\frac{c}{e^{kx}}

Step-by-step explanation:

\frac{\mathrm{d} y}{\mathrm{d} x} =-ky

\frac{\mathrm{d} y}{\mathrm{d} x} +ky=0

comparing with equation

\frac{\mathrm{d} y}{\mathrm{d} x} + Py=Q(x)

I.F.= e^{\int P dx}

I.F.= e^{\int k dx}

I.F.= e^{kx}

y=\frac{1}{I.F.} ( \int {Q(x)}  dx  +c)

y=\frac{1}{e^{kx}} ( \int {0}  dx  +c)

y=\frac{c}{e^{kx}}

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Explanation:


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The vertex of the parabola is the minimum or maximum on the parabola.


If the parabola open down then the vertex is a maximum, and if the parabola open upward the vertex is a minimum.


The sign of the coefficient of the quadratic term tells whether the parabola opens upward or downward.


When such coefficient is positive, the parabola opens upward (so it has a minimum); when the coefficient is negative the parabola opens downward (so it has a maximum).


Here the coefficient is positive (3), which tells that the vertex of the parabola is a miimum.


Then, finding the minimum value of the function is done by finding the vertex.

I will change the form of the function to the vertex form by completing squares:

Given: 3x² + 12x + 16

Group: (3x² + 12x) + 16
Common factor: 3 [x² + 4x ] + 16
Complete squares: 3[ ( x² + 4x + 4) - 4] + 16
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Distributive property: 3 (x + 2)² - 12 + 16
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Then the minimum value is 4 (when x = - 2).


2) The othe function is <span>g(x)= 2 *sin(x-pi)
</span>

The sine function goes from -1 to + 1, so the minimum value of sin(x - pi) is - 1.


When you multiply by 2, you just increased the amplitude of the function and obtain the new minimum value is 2 (-1) = - 2


Comparing the two minima, you have 4 vs - 2, and so the function g(x) has the smallest minimum y-value.

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