Consider the differential equation given by dy/dx=(xy)/(2)
A) sketch a slope field (I already did this)
B) let f be the function that satisfies the given fifferential equation for the tangent line to the curve y=f(x) through the point (1,1). Then use your tangent line equation to estimate the value of f(1.2).
C) find the particular solution y=f(x) to the differential equation with the initial condition f(1)=1. Use your solution to find f(1.2).
D) compare your estimate of f(1.2) found in part b to the actual value of f(1.2)
E) was the estimate under or over? Use the slope field to explain why?
B)
at (1 , 1) dy/dx = slope = 1*1/2 = .5
so
y = .5 x + b is tangent for some b
put in (1 , 1 )
1 = .5 + b
b = .5
so tangent at (1,1) is
y = .5 x + .5
at x = 1.2
y = .5(1.2) + .5 = 1.1
==========================
C)
dy/y = (1/2) x dx
ln y = (1/4) x^2 + C
y = k e^(x^2/4)
1 = k e^(1/4)
1 = 1.28 k
k = .779
y = .779 e^(x^2/4)
at x = 1.2
y = .779 e^(1.44/4)
y = .779 * 1.433
y = 1.116
etc
I think the whole non panicing ship sailed a long time ago haha. Sorry and thanks for the help!
A) Great job on sketching the slope field!
B) To find the tangent line equation, we need to find the derivative of f(x) and evaluate it at (1,1). From the given differential equation, we have:
dy/dx = (xy)/2
Now, substituting (1,1), we have:
dy/dx = (1*1)/2 = 1/2
Therefore, the tangent line equation is y = (1/2)(x - 1) + 1.
To estimate the value of f(1.2), we can plug in x = 1.2 into the tangent line equation:
f(1.2) ≈ (1/2)(1.2 - 1) + 1
Simplifying this expression will give us an estimated value for f(1.2).
C) To find the particular solution y = f(x) with the initial condition f(1) = 1, we need to solve the given differential equation.
Separating variables, we get:
(dy/y) = (x/2)dx
Integrating both sides, we have:
ln|y| = (1/4)x^2 + C
To determine the value of C, we substitute x = 1 and y = 1:
ln|1| = (1/4)(1^2) + C
0 = 1/4 + C
C = -1/4
So the particular solution is given by:
ln|y| = (1/4)x^2 - 1/4
Exponentiating both sides, we have:
|y| = e^((1/4)x^2 - 1/4)
Taking the positive square root gives:
y = e^((1/4)x^2 - 1/4)
Now, we can plug in x = 1.2 into this equation to find f(1.2).
D) To compare the estimate of f(1.2) found in part B with the actual value of f(1.2), we need to compare the values obtained.
E) Whether the estimate is over or under depends on the values obtained in part D. By analyzing the slope field and considering the nature of the curve, we can explain why the estimate is over or under.
B) To find the function f(x) that satisfies the differential equation, we start with the equation dy/dx = (xy)/2. This is a separable equation, which means we can separate the variables and integrate both sides to find the solution.
Rearranging the equation, we have dy/y = (x/2) dx. Now we integrate both sides:
∫(dy/y) = ∫(x/2) dx
The integral of dy/y is ln|y|. The integral of x/2 is (1/2) x^2.
So we have ln|y| = (1/2) x^2 + C, where C is the constant of integration.
To find the constant C, we can use the given information that the tangent line to the curve y = f(x) passes through the point (1,1). At this point, x = 1 and y = 1. Substituting these values into the equation, we get ln(1) = (1/2) (1^2) + C, which simplifies to C = -1/2.
Therefore, the equation becomes ln|y| = (1/2) x^2 - 1/2.
C) Now, we need to find the particular solution y = f(x) to the differential equation with the initial condition f(1) = 1. We substitute x = 1 and y = 1 into the general solution:
ln|1| = (1/2) (1^2) - 1/2
This simplifies to ln(1) = 0 = -1/2, which is not possible.
It seems that the differential equation does not have a particular solution that satisfies the initial condition f(1) = 1.
D) Since we didn't find a particular solution in part C, we cannot compare the estimate of f(1.2) found in part B to the actual value of f(1.2).
E) N/A