Differential Equation

The given differential equation is:
P^3 = e^y

To solve this differential equation, we need to find the solution for y in terms of P. This can be done by taking the natural logarithm of both sides of the equation.

Solution

1. Taking the natural logarithm of both sides:
ln(P^3) = ln(e^y)

2. Applying the logarithmic property:
3ln(P) = y

3. Rearranging the equation:
y = 3ln(P)

4. Therefore, the solution to the given differential equation is y = 3ln(P).

Explanation

To solve the given differential equation, we follow the following steps:

1. Taking the natural logarithm of both sides:
- This step is necessary to isolate the variable y on one side of the equation.
- The natural logarithm of P^3 is written as ln(P^3) and the natural logarithm of e^y is written as ln(e^y).

2. Applying the logarithmic property:
- The logarithmic property states that the logarithm of a power is equal to the product of the exponent and the logarithm of the base.
- In this case, the exponent of P^3 is 3, so the logarithm of P^3 is equal to 3 times the logarithm of P.
- Similarly, the exponent of e^y is y, so the logarithm of e^y is equal to y times the logarithm of e.
- Since the logarithm of e is equal to 1, the equation simplifies to 3ln(P) = y.

3. Rearranging the equation:
- To express y in terms of P, we rearrange the equation to isolate y on one side.
- Dividing both sides of the equation by 3 gives us y = 3ln(P).

4. Therefore, the solution to the given differential equation is y = 3ln(P).
- This equation represents the relationship between the variables y and P, where y is equal to 3 times the natural logarithm of P.
- The solution provides a functional form that describes the relationship between the variables in the given differential equation.