Solve the differential equation dy dx 9x y – Delving into the realm of differential equations, we embark on a journey to unravel the intricacies of dy/dx = 9x y. This equation, a cornerstone of mathematical modeling, holds profound significance in diverse scientific and engineering disciplines. Its solution unveils a treasure trove of insights, guiding us towards a deeper understanding of the dynamic world around us.
In this exploration, we will delve into the heart of this differential equation, unraveling its mysteries step by step. Along the way, we will uncover its method of solution, derive its general solution, and illuminate its significance. Furthermore, we will explore practical applications and real-world scenarios where this equation plays a pivotal role.
Differential Equation: dy/dx = 9x y
A differential equation is a mathematical equation that involves an unknown function and its derivatives. The given differential equation dy/dx = 9x y is a first-order linear differential equation, where y is the unknown function and x is the independent variable.
Method of Solution, Solve the differential equation dy dx 9x y
To solve the differential equation dy/dx = 9x y, we use the method of separation of variables. This method involves separating the variables y and x on different sides of the equation and then integrating both sides.
Solution to the Equation
Separating the variables, we get:
“`dy/y = 9x dx“`
Integrating both sides, we obtain:
“`ln|y| = 4.5x^2 + C“`
where C is the constant of integration.
Solving for y, we get the general solution:
“`y = Ce^(4.5x^2)“`
Examples and Applications
This differential equation is relevant in various real-world applications, such as:
- Modeling population growth, where y represents the population size and x represents time.
- Analyzing the decay of radioactive substances, where y represents the amount of radioactive material and x represents time.
- Studying the spread of infectious diseases, where y represents the number of infected individuals and x represents time.
Questions and Answers: Solve The Differential Equation Dy Dx 9x Y
What is the significance of solving differential equations?
Differential equations are essential tools for modeling and analyzing dynamic systems across a wide range of disciplines. Their solutions provide valuable insights into the behavior of these systems, enabling predictions, optimizations, and control strategies.
How does the method of separation of variables apply to solving dy/dx = 9x y?
The method of separation of variables involves expressing the equation in terms of two separate functions, one dependent on x and the other on y. This allows us to solve each function independently, leading to the final solution of the differential equation.
What are some real-world applications of differential equations?
Differential equations find applications in countless fields, including population modeling, heat transfer analysis, electrical circuit design, and chemical reaction kinetics. They provide a powerful framework for understanding and predicting the behavior of complex systems in diverse areas of science and engineering.
