Differential equations are fundamental in mathematics, describing the relationship between a function and its derivatives. First order differential equations specifically involve only the first derivative of a function. This section will provide a deeper understanding of separable, linear, and exact first order differential equations. For a broader perspective on differential equations, see the overview of first order differential equations.
Separable Differential Equations
Separable differential equations are those where terms involving the dependent variable can be separated from terms involving the independent variable. They can be represented as: dy/dx = f(y) * g(x)
- Characteristics:
- 1. They can be rearranged to have all terms involving y on one side and all terms involving x on the other.
- 2. Commonly used in problems related to growth and decay, similar to scenarios in free fall and projectile motion.
- Solution Method:
- 1. Separate the variables to get all y's on one side and all x's on the other.
- 2. Integrate both sides, applying basic integration techniques where necessary
- Example: For the equation dy/dx = yx. Separating variables, we get: dy/y = x dx Integrating both sides, the solution is: ln|y| = x2/2 + C
IB Maths Tutor Tip: Mastering first order differential equations involves recognising their type quickly. Practice identifying separable, linear, and exact equations to efficiently apply the appropriate solving technique.
Linear Differential Equations
Linear differential equations are of the first order and can be written in a standard linear form as: dy/dx + P(x)y = Q(x)
- Characteristics:
- 1. They are linear in y and its first derivative.
- 2. P(x) and Q(x) are functions of x alone.
- Solution Method:
- 1. Multiply the equation by an integrating factor, often denoted as u(x), a technique that can be further explored through the study of L'Hôpital's rule when dealing with limits and indeterminate forms.
- 2. The equation then becomes exact and can be integrated directly.
- Example: For the equation dy/dx + 2y = 3x. The integrating factor is e(2x). Multiplying through, we get: e(2x) dy/dx + 2e(2x)y = 3xe(2x) Integrating, the solution is: y = 1.5x - 0.25 + Ce(-2x)
Exact Differential Equations
Exact differential equations can be expressed as: M(x,y) + N(x,y) dy/dx = 0 where M and N are continuously differentiable functions of x and y.
- Characteristics:
- 1. The equation is 'exact' if the partial derivative of M with respect to y equals the partial derivative of N with respect to x.
- 2. If not exact, it can sometimes be made exact by multiplying with an integrating factor.
- Solution Method:
- 1. Verify if the equation is exact.
- 2. If exact, there's a function Psi(x,y) such that the partial derivative of Psi with respect to x equals M and with respect to y equals N.
- 3. Integrate to find Psi and equate it to a constant, a process that could be extended to second order differential equations for more complex scenarios.
IB Tutor Advice: When revising first order differential equations, focus on solving a variety of examples. This reinforces your understanding and improves your ability to tackle unexpected questions in exams.
Example: For the equation (3x2 + y2) + (2xy + x3) dy/dx = 0. The equation is exact. Integrating, the solution is: Psi(x,y) = x3y + y3/3 = C
FAQ
Yes, a non-exact differential equation can often be made exact by multiplying it with an appropriate integrating factor. This integrating factor is a function of x and y that, when multiplied with the original equation, makes it satisfy the condition for exactness. Once the equation becomes exact, it can be solved using the methods for exact differential equations. Finding the right integrating factor can sometimes be a challenge, but there are standard methods and formulas available for common types of non-exact equations to determine the suitable integrating factor.
A differential equation of the form M(x,y) + N(x,y) dy/dx = 0 is deemed exact if the partial derivative of M with respect to y equals the partial derivative of N with respect to x. In mathematical terms, if ∂M/∂y = ∂N/∂x, then the equation is exact. This condition ensures that there exists a function, often denoted as Psi(x,y), such that its partial derivatives with respect to x and y correspond to M and N, respectively. If a differential equation is exact, it means that we can find a potential function whose differential equals the given differential equation.
Absolutely! First-order differential equations are widely used in various fields to model real-world scenarios. In biology, they are used to model population growth, decay of substances, and the spread of diseases. In physics, they can describe motion, electrical circuits, and heat transfer, among other phenomena. In economics, they can represent rates of change in various economic factors. The ability of differential equations to describe the rate of change of one quantity in relation to another makes them invaluable in predicting and understanding dynamic systems and processes in the real world.
Separable differential equations are termed "separable" because they can be rearranged in such a way that all terms involving the dependent variable (often denoted as y) are separated from all terms involving the independent variable (often denoted as x). This separation allows us to integrate each side of the equation with respect to its respective variable. The ability to separate variables and integrate simplifies the process of finding the solution, making these equations one of the more straightforward types of differential equations to solve.
The integrating factor is a mathematical tool used to transform a non-exact differential equation into an exact one, making it easier to solve. When we multiply a linear differential equation by its integrating factor, the resulting equation can be expressed as the derivative of the product of the unknown function and the integrating factor. This allows us to integrate both sides directly, leading to the solution. The integrating factor is not unique; different factors can be used to solve the same equation. However, the most commonly used integrating factor is derived from the coefficient of the dependent variable in the differential equation.
Practice Questions
To solve the differential equation, we separate the variables to get all terms involving y on one side and all terms involving x on the other.
Rearranging, we have: dy/y = 2x dx
Now, we integrate both sides: Integrating dy/y with respect to y, we get ln|y|. Integrating 2x with respect to x, we get x2.
Thus, the solution is: ln|y| = x2 + C, where C is the constant of integration.
The given equation is in the standard form of a linear differential equation. To solve it, we can use an integrating factor. The integrating factor for this equation is e(3x).
Multiplying the entire equation by e(3x), we get: e(3x) dy/dx + 3e(3x)y = 6e(3x)
The left-hand side of this equation is the derivative of y times e(3x) with respect to x.
Integrating both sides with respect to x, we find: y times e(3x) = 2e(3x) + C
Where C is the constant of integration.
To find y, we divide both sides by e(3x): y = 2 + C times e(-3x)
Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.