Differentialekvationer separabla
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Section : Separable Equations
We are now going to start looking at nonlinear first order differential equations. The first type of nonlinear first order differential equations that we will look at is separable differential equations.
A separable differential equation is any differential equation that we can write in the following form.
\[\begin{equation}N\left( y \right)\frac{{dy}}{{dx}} = M\left( x \right)\label{eq:eq1} \end{equation}\]Note that in order for a differential equation to be separable all the \(y\)'s in the differential equation must be multiplied by the derivative and all the \(x\)'s in the differential equation must be on the other side of the equal sign.
To solve this differential equation we first
Differential equations/Separable differential equations
First Order Differential Equations
The order of a differential equation is the largest derivative involved. For example, in the equation , the largest derivative is the second, so the order is 2.
Separable Equations
[edit | edit source]One of the easiest class of ODEs to solve is separable equations.
Definition
[edit | edit source]A differential equation is called separable when it can be manipulated into an equation with the dependent variable and its differentials on one side of the equality, and the independent variable and its differentials on the other side. Thus, each side is in terms of a single variable.
Example
[edit | edit source]The equation has a fairly obvious solution if you know your differentiation rules well. Recall that and hence is a solution. But how could we have found this if we did not remember that happened to be its own derivative? Additionally, is it possible to find any more solutions? Observe that the equation above is separable, and can be written as . Now that both sides are in terms of their own variable, we can integrate:
And thus, . Since and are
Separable Differential Equations
FAQs on Separable Differential Equations
What are Separable Differential Equations in Calculus?
Differential equations in which the variables can be separated from each other are called separable differential equations. A general form to write separable differential equations is dy/dx = f(x) g(y), where the variables x and y can be separated from each other.
How to Identify Separable Differential Equations?
Any differential equation which can be written in any of the following forms is a separable differential equation:
- f(x) dx = g(y) dy
- dy/dx = f(x)/g(y)
- dy/dx = f(x) g(y)
- g(y) dy/dx = f(x)
How to Solve Separable Differential Equations?
To solve separable differential equations, we can follow the basic steps given below:
- Step 1: Write the derivative as a product of functions of individual variables, i.e., dy/dx = f(x) g(y)
- Step 2: Separate the variables bygd writing them on each side of the equality, i.e., dy/g(y) = f(x) dx
- Step 3: Integrate both sides and find the value of y, and hence the general solution of the separable differential equation, i.e., ∫ dy/g(y) = ∫ f(x) dx