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
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 dy/dx = 4y/x^2 with the initial condition y(-4) = e, we need to solve the differential equation and find the value of y.
Let's start by separating the variables. We can rewrite the differential equation as:
dy/y = 4*dx/x^2
Now, we integrate both sides with their respective variables to obtain:
∫(dy/y) = ∫(4*dx/x^2)
Integrating the left side with respect to y gives us:
ln|y| = -4/x + C1
Here, C1 is the constant of integration.
Next, we exponentiate both sides of the equation to eliminate the natural logarithm:
|y| = e^(-4/x + C1)
Since we want to find the specific solution, we can drop the absolute value signs and rewrite the equation as:
y = ±e^(-4/x + C1)
Now, we need to apply the initial condition y(-4) = e to determine the value of the constant C1.
Substituting x = -4 and y = e into the equation gives us:
e = ±e^(-4/(-4) + C1)
e = ±e^2 + C1
Simplifying further, we get:
e - ±e^2 = C1
Now, we can rewrite the specific solution using the determined constant value:
y = ±e^(-4/x + (e - ±e^2))
Therefore, the specific solution of the given differential equation with the condition y(-4) = e is represented by the equation:
y = ±e^(-4/x + (e - ±e^2))
dy/dx = 4y/x^2
dy/y = 4/x^2 dx
lny = -4/x + c
y = c e^(-4/x)
Now use the condition to find c