The rate of change of y with respect to x is one-half times the value of y. Find an equation for y, given that y = -7 when x = 0.
You get:
A. y=-7e^0.5x
B. y=e^0.5x -7
C. y=-7(1/2)^x
D. y=-7(1/2)e^x
E. dy/dx=1/2*y
It's A, y=-7e^0.5x
come on. it's very similar.
I see you didn't take my suggestion to use math instead of words, either.
dy/dx = y/2
dy/y = 1/2 dx
lny = x/2 + c
y = c*e^(x/2)
Now use y(0) = -7 to finish it off
and next time, try showing some of your work when asking for help.
Well, let's see. We know that the rate of change of y with respect to x is one-half times the value of y. So we can write that as:
dy/dx = (1/2) * y
Now, let's solve this differential equation. To do that, we can separate the variables and integrate both sides:
1/y dy = (1/2) dx
Integrating both sides gives us:
ln|y| = (1/2) x + C
where C is the constant of integration. Now, let's use the initial condition y = -7 when x = 0 to find the value of C. Plugging these values into the equation, we get:
ln|-7| = (1/2) * 0 + C
ln(7) = C
So the equation becomes:
ln|y| = (1/2) x + ln(7)
Now we can eliminate the absolute value by taking the exponential of both sides:
|y| = e^[(1/2) x + ln(7)]
Since we are given that y = -7 when x = 0, we know that y is negative, so we can drop the absolute value:
y = -e^[(1/2) x + ln(7)]
And there you have it! The equation for y is:
y = -e^[(1/2) x + ln(7)]
So the answer is option D. You're welcome!
To find an equation for y, given that the rate of change of y with respect to x is one-half times the value of y, we can use the differential equation:
dy/dx = (1/2)*y
This is a separable differential equation that can be solved using separation of variables.
Step 1: Rewrite the equation
dy/y = (1/2)*dx
Step 2: Integrate both sides
∫(dy/y) = ∫((1/2)*dx)
Step 3: Evaluate the integrals
ln|y| = (1/2)*x + C1
Using the fact that ln|e^a| = a, we can rewrite the equation as:
ln|y| = ln|e^(1/2*x + C1)|
Step 4: Apply the exponentiation property
|y| = |e^(1/2*x + C1)|
Step 5: Remove the absolute value signs
y = ±e^(1/2*x + C1)
Step 6: Simplify the expression
y = ±e^(1/2*x) * e^(C1)
Since e^(C1) is a constant, we can substitute it with C:
y = ±Ce^(1/2*x)
Now, using the initial condition y = -7 when x = 0, we can find the value of C.
Plugging in the values, we have:
-7 = ±Ce^(1/2*0)
-7 = ±C
Since C can be both positive and negative, we can rewrite the equation as:
y = Ce^(1/2*x)
So, the correct equation for y is given by option C. y = -7(1/2)^x.
To find an equation for y, we need to solve the given differential equation. The differential equation states that the rate of change of y with respect to x is one-half times the value of y.
We can start by separating variables. We rearrange the equation as follows:
dy/dx = (1/2) * y
Now, we can move the y term to one side and the dx term to the other side:
dy/y = (1/2) * dx
Next, we integrate both sides of the equation. On the left side, we integrate with respect to y and on the right side with respect to x:
∫(1/y) dy = ∫(1/2) dx
The integral of (1/y) with respect to y is ln|y| + C1, where C1 represents the constant of integration. Similarly, the integral of (1/2) with respect to x is (1/2)x + C2, where C2 represents another constant of integration.
Applying the integrals, we have:
ln|y| + C1 = (1/2)x + C2
We can combine the constants of integration into a single constant:
ln|y| = (1/2)x + C
Now, we can remove the logarithm by exponentiating both sides of the equation:
e^(ln|y|) = e^((1/2)x + C)
The exponential of the natural logarithm simplifies to the absolute value of y:
|y| = e^((1/2)x + C)
Finally, we can eliminate the absolute value by considering the initial condition given in the problem. When x = 0, y = -7. We substitute these values into the equation:
|-7| = e^((1/2)(0) + C)
7 = e^(C)
Now, we can rewrite the equation without the absolute value:
y = ±e^(1/2)x * e^C
Since the constant C can be any real number, including zero, we can simplify the equation to:
y = Ae^(1/2)x
Where A is a non-zero constant determined by the initial condition. In this case, y = -7 when x = 0, so we can substitute these values once again:
-7 = Ae^(1/2)(0)
-7 = A
Therefore, the equation for y is:
y = -7e^(1/2)x
Hence, the answer is A. y = -7e^(1/2)x.