Find the general solution to the differential equation that models the statement:"The rate of change of P with respect to x is proportional to 15-x.
dP/dx = k(15-x)
now take it from there...
k(15-x)
To find the general solution to the given differential equation, we can follow these steps:
Step 1: Write down the differential equation.
Let P be the function representing the quantity, and x be the independent variable. The differential equation is given as:
dP/dx = k(15 - x)
Step 2: Separate the variables.
We can rewrite the equation by separating the variables:
dP = k(15 - x) dx
Step 3: Integrate both sides.
We integrate both sides of the equation:
∫ dP = ∫ k(15 - x) dx
On the left side, the integral of dP is simply P. On the right side, we integrate k(15 - x) with respect to x:
P = -k∫ (x - 15) dx
Step 4: Evaluate the integral.
Integrating (x - 15) with respect to x gives us:
P = -k[(1/2)x^2 - 15x] + C
The constant C is the constant of integration, which represents the family of solutions for the given differential equation. Thus, the general solution to the differential equation is:
P = -k[(1/2)x^2 - 15x] + C
Where k is a constant and C represents the constant of integration.
To find the general solution to the given differential equation:
"The rate of change of P with respect to x is proportional to 15-x,"
we start by expressing this relationship mathematically. Let's denote the rate of change of P with respect to x as dP/dx. The given statement implies:
dP/dx = k*(15-x),
where k is the proportionality constant.
To solve this differential equation, we need to separate the variables and integrate both sides. Rearranging the equation, we get:
dP = k*(15-x)*dx.
Now, we can integrate both sides with respect to their respective variables. Integrating the left side with respect to P gives P. On the right side, we integrate with respect to x:
∫ dP = ∫ k*(15-x)*dx.
Integrating, we find:
P = k*(15x - (1/2)*x^2) + C,
where C is the constant of integration.
So, the general solution to the differential equation is:
P(x) = k*(15x - (1/2)*x^2) + C.