is dy/dx=5/x+y separable? If so, how?
Thank You.
Is the given equation
dy/dx= (5/x) + y (≡ dy/dx=5/x+y)
or
dy/dx=5/(x+y) ?
it is dy/dx= (5/x) + y
An equation has separable variables if it can be rearranged to be in the following form:
A(x)dx + B(y)dy = 0 ....(1)
For example,
dy/dx = -y/x can be rewritten as
dy/y = -dx/x
Integrate both sides,
ln(y)=-ln(x)+C1
or
y=C/x where C is a constant.
The given equation cannot be rearranged to the form of equation (1), so it is not separable.
However, it can be solved with the integrating factor e-x, resulting in
y=ex*∫5e-xdx/x
thanks =]
Yes, the differential equation dy/dx = 5/(x+y) is separable. To solve a separable differential equation, you want to separate the variables and then integrate both sides.
To start, let's rewrite the given equation in a more convenient form:
dy/(dx) = 5/(x+y)
Now, we can multiply both sides by dx to separate the variables:
dy = (5/(x+y)) dx
Next, we want to rearrange the equation so that all the y terms are on one side and all the x terms are on the other side. In this case, we can bring dx to the right side and y terms to the left side:
dy = 5/(x+y) dx
dy = 5dx/(x+y)
Now, we can separate the variables by multiplying both sides by (x+y):
(x+y) dy = 5dx
The variables are now separated. Next, we integrate both sides of the equation with respect to their respective variables. Integrating the left side with respect to y and the right side with respect to x, we get:
∫(x+y) dy = ∫5dx
Integrating the left side:
∫(x+y) dy = ∫(x+1)y dy
= (1/2)(x^2 + 2x + y^2) + C1 (where C1 is the constant of integration)
Integrating the right side:
∫5dx = 5x + C2 (where C2 is the constant of integration)
Now, our equation becomes:
(1/2)(x^2 + 2x + y^2) + C1 = 5x + C2
Simplifying the equation further, we have:
1/2x^2 + x + 1/2y^2 + C1 = 5x + C2
We can combine the constants of integration into a single constant, so the final solution is:
1/2x^2 + x + 1/2y^2 = 5x + C (where C = C1 - C2)
Therefore, the separable solution for the given differential equation is 1/2x^2 + x + 1/2y^2 = 5x + C.