Question

eliminate the arbitrary constant from y= Ae^2x +Be^x + C

83

likes
415 views

Answer to a math question eliminate the arbitrary constant from y= Ae^2x +Be^x + C

Expert avatar
Maude
4.7
58 Answers
To eliminate the arbitrary constant from the given equation, we need to find the relationship between A, B, and C. We can do this by differentiating the equation with respect to x.

Step 1: Differentiate the equation y = Ae^(2x) + Be^x + C with respect to x.

\begin{align*}
\frac{dy}{dx} &= \frac{d}{dx}(Ae^(2x)) + \frac{d}{dx}(Be^x) + \frac{d}{dx}(C) \
&= 2Ae^(2x) + Be^x + 0 \
\frac{dy}{dx} &= 2Ae^(2x) + Be^x
\end{align*}

Step 2: Now, equate this differentiated equation to zero to find the relationship between A, B, and C.

2Ae^(2x) + Be^x = 0

Step 3: Factor out the common exponential term e^x.

e^x (2Ae^x + B) = 0

Step 4: Since e^x is always positive and never equal to zero, we can set the expression within parentheses equal to zero.

2Ae^x + B = 0

Step 5: Solve this equation for A in terms of B.

2Ae^x = -B

Ae^x = -B/2

A = -\frac{B}{2e^x}

Step 6: Substitute the expression for A back into the original equation y = Ae^(2x) + Be^x + C.

y = (-\frac{B}{2e^x})e^(2x) + Be^x + C

y = -\frac{Be^x}{2} + Be^x + C

Step 7: Simplify the expression by finding a common denominator.

y = \frac{2Be^x - Be^x}{2} + C

y = \frac{Be^x}{2} + C

Answer: The equation without the arbitrary constant is y = \frac{Be^x}{2} + C.

Frequently asked questions (FAQs)
What is the volume of a cone with a radius of 5 inches and a height of 10 inches?
+
Question: Find the exact values of x, where 0 < x < 2π, such that cot x = -√3.
+
Math question: What is the median of the numbers 3, 5, 7, 11, and 15?
+
New questions in Mathematics
Find 2 numbers that the sum of 1/3 of the first plus 1/5 of the second will be equal to 13 and that if you multiply the first by 5 and the second by 7 you get 247 as the sum of the two products with replacement solution
1/2x +3 <4x-7
4.2x10^_6 convert to standard notation
Two numbers differ by 7, and the sum of their squares is 29. Find the numbers.
A National Solidarity Bond offers A 5 year bond offering a gross return of 15% Calculate the AER for this investment. (Give your answer to two decimal places, no need for the percent or € sign in your answer)
4. Show that if n is any integer, then n^2 3n 5 is an odd integer
There are four times as many roses as tulips in Claire’s garden. Claire picked half of the number of roses and 140 roses were left in the garden. How many roses and tulips were in the Garden the first?
find f(x) for f'(x)=3x+7
User Before the election, a poll of 60 voters found the proportion who support the Green candidate to be 25%. Calculate the 90% confidence interval for the population parameter. (Give your answers as a PERCENTAGE rounded to TWO DECIMAL PLACES: exclude any trailing zeros and DO NOT INSERT THE % SIGN) Give the lower limit of the 90% confidence interval Give the upper limit of the 90% confidence interval
Shows two blocks, masses 4.3 kg and 5.4 kg, being pushed across a frictionless surface by a 22.5-N horizontal force applied to the 4.3-kg block. A. What is the acceleration of the blocks? B. What is the force of the 4.3-kg block on the 5.4 -kg block? C. What is the force of the 5.4 -kg block on the 4.3 -kg block?
User One of the applications of the derivative of a function is its use in Physics, where a function that at every instant t associates the number s(t), this function s is called the clockwise function of the movement. By deriving the time function we obtain the velocity function at time t, denoted by v(t). A body has a time function that determines its position in meters at time t as S(t)=t.³√t+2.t . Present the speed of this body at time t = 8 s.
Find the complement and supplement angles of 73
A contractor gives a bank note for $10250 at a rate of 1% for one month. How much interest is charged for 4 months?
To get to a hotel on the hill you have to travel 6 km of uphill road and every kilometer there are 6 sharp curves. Each of the sharp curves is marked by three traffic signs. How many traffic signs are there on the stretch of road that leads to the arbergi?
A person runs 175 yards per minute write a variable that represents the relationship between time and distance
The mean of 4 numbers is 5 and the mean of 3 different numbers is 12. What is the mean of the 7 numbers together? Produce an algebraic solution. Guess and check is acceptable.
2x-4=8
Determine the general solution of the equation y′+y=e−x .
Find the rule that connects the first number to the second number of each pair. Apply the rule to find the missing number in the third pair. (18 is to 22) (54 is to 26) (9 is to ?)
5 1/9 + 2 2/3