๐ Definition of Separable Variables in Differential Equations
A separable differential equation is one that can be written in the form:
$\frac{dy}{dx} = f(x)g(y)$
Where $f(x)$ is a function of $x$ only, and $g(y)$ is a function of $y$ only. The key is that you can separate the variables $x$ and $y$ to different sides of the equation before integrating.
๐ History and Background
The concept of separable differential equations emerged alongside the development of calculus in the 17th century. Mathematicians like Leibniz and Bernoulli explored methods for solving these types of equations, laying the foundation for more advanced techniques in differential equations. The ability to separate variables greatly simplified the process of finding solutions, making it a fundamental tool in various scientific disciplines.
๐ Key Principles
- ๐ Separation: The primary goal is to rewrite the equation so that all terms involving $y$ (including $dy$) are on one side, and all terms involving $x$ (including $dx$) are on the other side.
- ๐ก Integration: After separating the variables, integrate both sides of the equation with respect to their respective variables.
- ๐ Solving: Solve the resulting equation for $y$ as a function of $x$ (or vice versa) to obtain the general solution.
- โ Constant of Integration: Don't forget to include the constant of integration ($C$) after integrating both sides.
- ๐ฑ Initial Conditions: If initial conditions are given, use them to solve for the constant of integration and find the particular solution.
๐งฎ Steps to Solve Separable Differential Equations
- โ๏ธ Step 1: Rewrite the equation: Express the differential equation in the form $\frac{dy}{dx} = f(x)g(y)$.
- โ Step 2: Separate the variables: Divide or multiply to get all $y$ terms with $dy$ on one side, and all $x$ terms with $dx$ on the other side, resulting in an equation like $\frac{dy}{g(y)} = f(x) dx$.
- โซ Step 3: Integrate both sides: Integrate both sides of the equation with respect to their respective variables: $\int \frac{dy}{g(y)} = \int f(x) dx$.
- โ
Step 4: Solve for y: Solve the resulting equation for $y$ in terms of $x$. This may involve algebraic manipulation.
- โ Step 5: Add the constant of integration: Add the constant of integration to one side of the equation.
- ๐ฏ Step 6: Apply initial conditions (if given): If you have an initial condition, such as $y(x_0) = y_0$, plug these values into the solution to find the specific value of the constant of integration.
- โจ Step 7: Write the final solution: Write the final solution with the constant of integration. If the initial condition was given solve and plug in.
๐ Real-world Examples
- ๐ก๏ธ Newton's Law of Cooling: Describes how the temperature of an object changes over time in relation to its surroundings. The differential equation is separable.
- ๐ฑ Population Growth: Models population growth rates, often assuming the growth rate is proportional to the current population.
- โข๏ธ Radioactive Decay: Describes the decay of radioactive substances, where the rate of decay is proportional to the amount of substance present.
- ๐ง Mixing Problems: Used to model the amount of a substance in a tank as a mixture is added and drained.
๐งช Example Problem
Solve the differential equation $\frac{dy}{dx} = x y$ with the initial condition $y(0) = 2$.
- Separate variables: $\frac{dy}{y} = x dx$
- Integrate both sides: $\int \frac{dy}{y} = \int x dx$ which gives $\ln|y| = \frac{x^2}{2} + C$
- Solve for y: $y = e^{\frac{x^2}{2} + C} = e^{\frac{x^2}{2}}e^C = Ae^{\frac{x^2}{2}}$, where $A = e^C$
- Apply initial condition: $2 = Ae^{\frac{0^2}{2}} = A$, so $A = 2$
- Final solution: $y = 2e^{\frac{x^2}{2}}$
๐ Conclusion
Separable differential equations are a fundamental type of differential equation that can be solved by separating the variables and integrating. They appear in numerous applications across various scientific and engineering fields. Understanding the principles and techniques for solving these equations is crucial for any student studying differential equations.