Write and then solve for y = f(x) the differential equation for the statement: "The rate of change of y with respect to x is inversely proportional to y^4."
dy/dx = k / (y^4) where k is a constant
Separate the variables
(y^4) dy = k dx
Integrate both sides
(y^5) / 5 = kx + c where c is also a constant
To write the differential equation for the given statement, we can use the following information:
"The rate of change of y with respect to x" can be written as dy/dx.
"The rate of change of y with respect to x is inversely proportional to y^4" can be written as:
dy/dx ∝ 1/y^4.
To incorporate the proportionality constant, let's introduce a constant k:
dy/dx = k/y^4.
This is the differential equation that represents the given statement. To solve this differential equation, we can separate the variables and then integrate.
Separating the variables:
dy/y^4 = k*dx.
Integration:
∫(1/y^4) dy = ∫k*dx.
To integrate 1/y^4, we can rewrite it as y^(-4) and use the power rule of integration:
∫y^(-4) dy = ∫k*dx.
This becomes:
-1/3 y^(-3) = kx + C,
where C is the constant of integration.
Now, let's solve for y:
-1/3 y^(-3) = kx + C.
Multiplying both sides by -3:
y^(-3) = -3kx - 3C.
Taking the reciprocal of both sides:
y^3 = (-1/3)(1/(-3kx - 3C)).
Simplifying:
y^3 = 1/(9kx + 9C).
Finally, taking the cube root of both sides:
y = (1/(9kx + 9C))^(1/3).
This is the solution to the differential equation y' = k/y^4 representing the given statement.