# Help with this separable differential equation

• dmayers94
In summary, the conversation discusses a problem of dx/dt = (x+9)^2, which is separable. The person mentions trying to use partial fractions, but it does not work. Another person suggests using the power rule for integration as an alternative method.
dmayers94
The problem is dx/dt = (x+9)^2.
This is separable so I made it dx / (x+9)^2 = dt.
The only method I can think of using for something like this is partial fraction, but I can't get it to work with A/(x+9) + B/(x+9).
Can anyone find a method that works?

dmayers94 said:
The problem is dx/dt = (x+9)^2.
This is separable so I made it dx / (x+9)^2 = dt.
The only method I can think of using for something like this is partial fraction, but I can't get it to work with A/(x+9) + B/(x+9).
Can anyone find a method that works?

FYI, the partial fraction decomposition would be $\frac{A}{(x+9)}+\frac{B}{(x+9)^2}$.

This aside, you can simply use the idea of the power rule for integration. What is the integral of 1/u2, for example?

## 1. What is a separable differential equation?

A separable differential equation is a type of differential equation where the variables can be separated and solved independently. This means that the differential equation can be written in the form of dy/dx = f(x)g(y), where f(x) is a function of x and g(y) is a function of y.

## 2. How do I solve a separable differential equation?

To solve a separable differential equation, you can follow these steps:
1. Write the differential equation in the form dy/dx = f(x)g(y).
2. Separate the variables by moving all terms with y to the left side and all terms with x to the right side.
3. Integrate both sides with respect to their respective variables.
4. Solve for y to get the general solution.
5. If initial conditions are given, use them to find the particular solution.

## 3. What are some common techniques used to solve separable differential equations?

Some common techniques used to solve separable differential equations include:
1. Separation of variables
2. Integrating factors
3. Exact differential equations
4. Substitution
5. Series solutions

## 4. What are some real-life applications of separable differential equations?

Separable differential equations have various real-life applications, such as:
1. Population growth models
3. Cooling or heating processes
4. Chemical reactions
5. Electrical circuits

## 5. Are there any special cases of separable differential equations?

Yes, there are some special cases of separable differential equations, such as:
1. Homogeneous differential equations
2. Bernoulli differential equations
3. Linear differential equations
4. Autonomous differential equations
5. Non-linear differential equations

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