Profile
Settings
Refer your friends
Sign out
Introduction
Across various science and engineering problems, we are required to solve differential equations. The logic of solving differential equations is quite established now, and we have many theoretical ways to solve them. Nowadays, scientists have come up with various ordinary differential equation solvers to solve them computationally. For differential equations in which the variables can be separated, we can find solutions using separable kernels. These can then be employed to foe the modeling and predicting the variation of various physical quantities such as pressure, temperature, current, etc. Ordinary differential equations are also solved to model simple and complex biological pathways. Thus we can see that understanding the mathematics of differential equations has led to their applications in various fields.
According to algebra, the Kernel and differential equations are generally the inverse image of zero. The Kernel of a matrix, similarly known as null space, is the Kernel of the linear map defined by the matrix. It helps us to define quotient objects while also being utilized in structures.
The differential equation can be defined as an equation containing one or more function derivatives. Differential equations can be classified based on:
- Partial differential equations: It involves partial derivatives.
- Ordinary differential equations: It does not involve partial derivatives. Further ordinary differential equations can be further classified based on the order. The order of a differential equation can be defined as the highest derivative that occurs in an equation. For example:
This is an example of a second-order differential equation as it has a double derivative denoted by d2y/dx2. Further, we can classify them as being linear and non-linear. Now we will learn what separable differential equations are and how to solve them.
Separable Differential Equation:
Separable Differential Equations are a particular class of differential equations. These differential equations can be solved by separating the variables in the equation. Separable Differential Equations can be represented in the form dy/dx = f(x) g(y), where variables x and y are separate. This means that the equation can be written explicitly as the variables of x and y. Some other forms of separable differential equations are:
f(x) dx = g(y) dy
dy/dx = f(x)/g(y)
dy/dx = f(x) g(y)
g(y) dy/dx = f(x)
These forms can be identified as separable differential equations and can be solved using the separation of variables.
Variable Separable Method:
The first step to solving separable differential equations is to identify if they suit the format of separable differential equations. Then the next step is to use the method of separation of variables. By separating the variables in the equation,
dy/dx = f(x) g(y),
We get the resulting equation as
dy/g(y) = f(x) dx
Then integrate both sides of the equation, i.e.,
∫ dy/g(y) = ∫ f(x) dx
And solve for the value of y and to find the solution of the separable differential equation. This is the variable separable method.
Applications of solving differential equations: The various applications of differential equations and solutions with separable kernels can be listed as:
- Population Models: A practical application of differential equations is its use in population models. The Malthusian law of population growth dp/dt = RP represents the change in population with time. The value of r is a constant that is different for different species. This law predicts the growth of a species concerning time and is highly influential in microbial kinetic growth etc.
- In studying transient and steady-state in an electric circuit, differential equations are applied. These differential equations that describe the circuit will have two distinctions to its solutions. The complementary function represents the transient state, and the solution corresponds to the steady-state.
- Differential equations can also be used to solve exponential growth or decay. E.g.:
dx/dt=kx
Here x and t are variables. The variable represents time, and ‘k’ is a constant. The variable ‘x’ is a function of time. When the value of ‘k’ is positive, there is a continuous growth rate, and if ‘k’ is harmful, there is constant decay. This is known as death kinetics and is reciprocal of the above-mentioned growth kinetics.
Conclusion
Thus, during the article, we have understood how to get solutions with separable kernels. We further studied kernel and differential equations’ definitions while also exploring its types and how to classify them. Then we had a brief overview of separable differential equations and their mathematical format and subsequently found out how to solve them. Finally, we must pair up theoretical knowledge with some practical applications. To achieve this, we had an overview of the various applications of differential equations in our daily lives and their solutions with separable kernels.
