Differential Equations

Step 1: Identifying the type of differential equation
The given equation is a first-order separable differential equation:
Rearrange to separate variables:
Step 2: Integrating both sides Integrating both sides:
Using the standard integral formula , we obtain:
Step 3: Solving for Exponentiating both sides: Let (a constant),
Step 4: Applying Initial Condition Given , substitute and :
This leads to a contradiction, meaning no solution satisfies the given initial condition.
Thus, the number of solutions is one, confirming that the given condition uniquely determines .

To find the differential equation involving the variables x and y, we can express y in terms of x and then differentiate it. Let's proceed with the given equations:

x = log(p) (Equation 1) y = 1/p (Equation 2)

First, let's solve Equation 1 for p: x = log(p) p = e^x (where e is the base of the natural logarithm) Now, substitute the value of p in terms of x into Equation 2: y = 1/p y = 1/(e^x)

To find the differential equation, we differentiate Equation 2 with respect to x:

dy/dx = d/dx (1/(e^x)) Using the chain rule: dy/dx = -1/(e^x) * d/dx(x)

Since d/dx(x) is equal to 1, we have: dy/dx = -1/(e^x)

So, the differential equation involving the variables x and y is: dy/dx = -1/(e^x)