# Separable Differential Equations Using Initial Values

• alexsylvanus
In summary: The speaker suggests that this method may be applicable for all separable differential equations, but is unsure and seeks verification. Additionally, the speaker mentions that vectorial arrows should not be included in the separated form of the equation.
alexsylvanus
AP Physics student here, I'm working on a problem that takes into account air resistance, where something is thrown up at initial velocity v_0, and the drag force is proportional to the velocity, so, $$\vec{F_{drag}}=-k\vec{v}$$.
Using Newtons second law and making up positive, down negative, you get, $$m\vec{a}=-k\vec{v}-mg$$
Which yields the differential equation, $$m\frac{d\vec{v}}{dt}=-k\vec{v}-mg$$
This can be separated to become, $$\frac{1}{k\vec{v}+mg}d\vec{v}=-\frac{1}{m}dt$$.
Now, when solving this, many sources say to use an indefinite integral on both sides, which results in a constant of integration, which you can then find by inputting t=0 and v=v_0. However, I thought that instead of using an indefinite integral and going to the trouble of finding the constant you could use a definite integral like so,
$$\int_{\vec{v_0}}^{\vec{v}}\frac{1}{k\vec{v}+mg}d\vec{v}=\int_{0}^{t}-\frac{1}{m}dt$$.
I worked through a few problems using my method, and it seemed to yield the same solutions as indefinite integration. Now, I was wondering if integrating from the known value (ex. v_0=v(t=0)) to the variable (v(t)) on one side of the differential equation and integrating from t=0 to t on the other side could possible work for all separable differential equations so that finding the constant of integration is no longer necessary. For example, if you know that y is a function of x and y(2)=6 for the differential equation, $$\frac{dy}{dx}=xy^2$$,
could you say that, $$\int_{6}^{y}\frac{1}{y^2}dy=\int_{2}^{x}xdx$$
My calculus teacher wasn't sure, so if anyone could verify that this method is or is not acceptable that would be greatly appreciated.

Acceptable.

But, lose the arrows over the a and v, or include an arrow over the g. Then lose all three arrows, and work only with the magnitudes of the vectors. You should not have vectorial arrows in the separated form of the equation, which should only involve scalar variables.

Chet

## What is a separable differential equation?

A separable differential equation is a type of differential equation where the variables can be separated into two distinct functions, making it easier to solve.

## How do I solve a separable differential equation?

To solve a separable differential equation, you need to separate the variables into two functions, then integrate both sides of the equation. This will give you an equation with just one variable, which you can then solve for.

## What are initial values in a separable differential equation?

Initial values refer to the conditions at the starting point of the differential equation. These conditions are usually given as specific values for the variables, which can help you solve the equation.

## Why do we use initial values in solving separable differential equations?

Initial values are used to help us find the particular solution to a separable differential equation. They provide a starting point for the equation and help us determine the constants of integration.

## Can separable differential equations be applied to real-world problems?

Yes, separable differential equations can be applied to various real-world problems, such as population growth, radioactive decay, and drug concentration in the body. They are a powerful tool for modeling and predicting the behavior of many natural phenomena.

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