# Calculus: separable differential equations

• alexis36
In summary: Once you have c, you "solve" for x.In summary, the conversation is about solving a separable differential equation for the growth rate of a tumor as it decreases in size. The solution involves separating variables, integrating both sides, and using the initial condition to find a constant.
alexis36

## Homework Statement

The question is discussing the growth rate of a tumor as it decreases in size (called the Gompertz equation: I am needing so SOLVE THE SEPERABLE DIFFERENTIAL EQUATION.

## Homework Equations

dx/dt = f(t)g(x) =(e^-t)(x(t)) x(o)=1

## The Attempt at a Solution

1. I am separating variables, (i think I am making a mistake here)
dx/dt=(e^-t)x(t)
dx/x(t)=(e^-t)(dt)
.. is that step right?

2. then i integrate both sides.

integrating 1/x(t) on one side.. and integrating e^-t(dt) on the other.. i get
ln|x|= -e^-t + c
.. now I am unsure as to how to solve the rest. am i solving for a variable for c?

alexis36 said:

## Homework Statement

The question is discussing the growth rate of a tumor as it decreases in size (called the Gompertz equation: I am needing so SOLVE THE SEPERABLE DIFFERENTIAL EQUATION.

## Homework Equations

dx/dt = f(t)g(x) =(e^-t)(x(t)) x(o)=1

## The Attempt at a Solution

1. I am separating variables, (i think I am making a mistake here)
dx/dt=(e^-t)x(t)
dx/x(t)=(e^-t)(dt)
.. is that step right?

2. then i integrate both sides.

integrating 1/x(t) on one side.. and integrating e^-t(dt) on the other.. i get
ln|x|= -e^-t + c
.. now I am unsure as to how to solve the rest. am i solving for a variable for c?
Looks to me like you have done everything right. Now use the fact that x(0)= 1 to find c. ln|1|= 0= -e^0+ c or 0= -1+ c. What is c? (c isn't, technically, a "variable", it is constant.)

## 1. What is a separable differential equation?

A separable differential equation is a type of differential equation that can be written in the form of y' = g(x)h(y), where y' represents the derivative of y with respect to x. This means that the variables x and y can be separated on opposite sides of the equation, making it easier to solve.

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

To solve a separable differential equation, you need to follow these steps:

1. Separate the variables x and y on opposite sides of the equation.

2. Integrate both sides of the equation with respect to x and y separately.

3. Solve for the constant of integration.

4. Rearrange the equation to solve for y.

5. Check your solution by plugging it into the original equation.

## 3. What is the importance of separable differential equations in calculus?

Separable differential equations are important in calculus because they are one of the most commonly used types of differential equations. They also serve as a foundation for other more complex types of differential equations, making it essential to understand how to solve them.

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

Some real-life applications of separable differential equations include population growth, chemical reactions, and radioactive decay. These equations can also be used to model the motion of objects, such as a falling object under the influence of gravity.

## 5. Are there any techniques to make solving separable differential equations easier?

Yes, there are some techniques that can make solving separable differential equations easier, such as using substitution or recognizing patterns in the equation. It is also helpful to practice solving various types of separable differential equations to become more familiar with the process.

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