Find the specific solution of the differential equation dy/dx = 4y/x^2 with condition y(−4) = e.
y equals negative 1 minus 4 divided by x <- my answer
y equals negative 1 times e raised to the 1 over x power
y equals e raised to the negative 4 over x power
None of these
To find the specific solution of the given differential equation, we'll need to separate the variables and integrate both sides.
Given the differential equation: dy/dx = (4y)/(x^2)
First, let's separate the variables by multiplying both sides of the equation by x^2 to get:
x^2 * dy/dx = 4y
Next, let's rewrite the equation as follows:
x^2 * dy = 4y * dx
Now, let's integrate both sides separately. On the left side, we integrate with respect to y, and on the right side, with respect to x.
∫ x^2 * dy = ∫ 4y * dx
∫ x^2 * dy = 4 ∫ y * dx
Integrating both sides gives us:
(x^2/2)y + C1 = 4xy + C2
Where C1 and C2 are constants of integration.
Now, let's apply the initial condition y(-4) = e.
Substituting x = -4 and y = e into the equation:
(-4^2/2)e + C1 = 4(-4)e + C2
16e + C1 = -16e + C2
Now, to find the specific values of C1 and C2, we can rearrange the equation:
C1 - C2 = -32e
Since C1 and C2 are constants, we can denote their difference as another constant, say C3, giving us:
C3 = -32e
Finally, substituting C3 = -32e back into the equation, we have:
(x^2/2)y = 4xy + C3
Simplifying the equation gives us the specific solution:
y = (4xy + C3)/(x^2/2)
In this case, none of the answer choices you provided (negative 1 minus 4 divided by x, negative 1 times e raised to the 1 over x power, or e raised to the negative 4 over x power) are the correct specific solution to the given differential equation with the given initial condition.