Solving Exponential equation 7^7x+5 = 2^x-1

All of these ODEs model a system with a spring, mass and dashpot.

Wewaii [24]

Answer:

− 3 y ' ' − 3 y ' + 3 y = 0 : over-damped

− 2 y ' ' − 4 y ' + 1 y = 0 : over-damped

1 y ' ' + 7 y ' + 5 y = 0: over-damped

Step-by-step explanation:

Using the characteristic equation you can express a differential equation of order n as an algebraic equation of degree n:

$a_ny^n+a_n_-_1y^{n-1}+...+a_1y'+a_oy=0$

This differential equation will have a characteristic equation of the form:

$a_nr^n+a_n_-_1r^{n-1}+...+a_1r+a_o=0$

Now, you can classify the solution for a differential equation using a simple method. In order to do it, you just need to use the discriminant.

If the discriminant is greater than zero, the solution is over-damped

If the discriminant is less than zero, the solution is under-damped

If the discriminant is equal to zero, the solution is critically damped

So, given the differential equation:

$-3y''-3y+3y=0$

Which has characteristic equation of the form:

$-3r^2-3r+3=0$

The quadratic polynomial of the form:

$ar^2+br+c=0$

Has discriminant:

$Disc=b^2-4ac$

In this case:

$a=-3\\b=-3\\c=3$

So:

$Disc=(-3)^2-4(-3)(3)=9-(-36)=45$

In this case:

$Disc=45>0$

Therefore the solution is over-damped.

Now, given the differential equation:

$-2y''-4y'+1y=0$

Which has characteristic equation of the form:

$-2r^2-4r+1=0$

The quadratic polynomial of the form:

$ar^2+br+c=0$

Has discriminant:

$Disc=b^2-4ac$

In this case:

$a=-2\\b=-4\\c=1$

So:

$Disc=(-4)^2-4(-2)(1)=16+8=24$

In this case:

$Disc=24>0$

Therefore the solution is over-damped.

Finally, given the differential equation:

$1y''+7y'+5y=0$

Which has characteristic equation of the form:

$1r^2+7r+5=0$

The quadratic polynomial of the form:

$ar^2+br+c=0$

Has discriminant:

$Disc=b^2-4ac$

In this case:

$a=1\\b=7\\c=5$

So:

$Disc=(7)^2-4(1)(5)=49-20=29$

In this case:

$Disc=29>0$

Therefore the solution is over-damped.

6 0

3 years ago

Solve 2x+3/2=14/3 What does x equal?

valina [46]

X= 19/12 which is 1.58 in decimal form

8 0

3 years ago

Write th number of permutations in factorial form. then simplify. How many different ways can you and six of your friends sit in

kolbaska11 [484]

The answer is D. The answer is D because when It is simplified 7?49. The answer is the only possible answer could be correct since the rest of their maths don't add up.
C
A 
C 
D 
D 
B 
D 
B 
C

8 0

4 years ago

Do any numbers multiply into 7 and add up to 8? or is it unfactorable

faltersainse [42]

1
1x7=7
1+7=8
1 is the answer

5 0

3 years ago

Please help with 11c!?

Contact [7]

Answer:

"To the nearest year, it would be about 9 years"

Step-by-step explanation:

11c)

This is compound growth problem. It goes by the formula:

$F=P(1+r)^t$

Where

F is the future amount

P is the present (initial) amount

r is the rate of growth, in decimal

t is the time in years

Given,

P = 20,000

r = 8% = 8/100 = 0.08

F = double of initial amount = 2 * 20,000 = 40,000

We need to find t:

$F=P(1+r)^t\\40,000=20,000(1+0.08)^t\\2=(1.08)^t$

To solve exponentials, we can take Natural Log (Ln) of both sides:

$2=(1.08)^t\\Ln(2)=Ln((1.08)^t)$

Using the rule shown below we can simplify and solve:

$Ln(a^b)=bLn(a)$

We can write:

$Ln(2)=Ln((1.08)^t)\\Ln(2)=tLn(1.08)\\t=\frac{Ln(2)}{Ln(1.08)}\\t=9.0064$

To the nearest year, that would be about 9 years

5 0

3 years ago