Solving The System Of Differential Equations Dx/dt = -2x And Dy/dt = -3x + Y
In the realm of differential equations, systems of equations often arise, demanding methods to unravel their intertwined solutions. This article delves into the systematic approach to solve a specific system of differential equations. Differential equations play a crucial role in modeling various phenomena in physics, engineering, economics, and other fields. A system of differential equations involves multiple equations with multiple unknown functions and their derivatives. Solving such a system means finding the functions that satisfy all the equations simultaneously. The system we aim to solve is defined as follows:
dx/dt = -2x
dy/dt = -3x + y
Where x and y are functions of the independent variable t. Our primary goal is to determine the expressions for x(t) and y(t) that satisfy both equations. The approach involves solving the first equation independently and then substituting the solution into the second equation. This transforms the second equation into a first-order linear differential equation, which can be solved using an integrating factor. Finally, we will obtain the general solutions for both x(t) and y(t), including arbitrary constants. These constants can be determined if initial conditions are provided. This article will demonstrate the step-by-step process, providing a clear and comprehensive understanding of the solution.
Our initial focus is on the first equation, which is a separable differential equation. This equation is expressed as:
dx/dt = -2x
To solve this equation, we employ the method of separation of variables, a fundamental technique for solving first-order differential equations. The essence of this method lies in isolating the variables on opposite sides of the equation. We rewrite the equation by dividing both sides by x and multiplying both sides by dt, yielding:
dx/x = -2 dt
Now that the variables are separated, we integrate both sides of the equation. The integral of dx/x with respect to x is the natural logarithm of the absolute value of x, denoted as ln|x|. The integral of -2 dt with respect to t is -2t. Thus, we have:
∫(dx/x) = ∫(-2 dt)
ln|x| = -2t + C₁
Here, C₁ represents the constant of integration. To solve for x, we exponentiate both sides of the equation using the exponential function (e). This gives us:
|x| = e^(-2t + C₁)
|x| = e^(-2t) * e^(C₁)
Since e^(C₁) is also a constant, we can replace it with another constant, say A. Moreover, we can remove the absolute value by introducing a ± sign. Thus, we obtain:
x(t) = ±A * e^(-2t)
We can absorb the ± sign into the constant A, allowing A to be any real number. Therefore, the general solution for x(t) is:
x(t) = A * e^(-2t)
This solution indicates that x(t) decays exponentially with time, with the rate of decay determined by the constant -2. The constant A determines the initial value of x at t = 0. This solution will be crucial in solving the second differential equation.
Having found the solution for x(t), we now turn our attention to the second differential equation:
dy/dt = -3x + y
We substitute the solution x(t) = A * e^(-2t) into this equation, which yields:
dy/dt = -3(A * e^(-2t)) + y
dy/dt = -3A * e^(-2t) + y
This is a first-order linear differential equation in the form dy/dt + P(t)y = Q(t), where P(t) = -1 and Q(t) = -3A * e^(-2t). To solve this type of equation, we use the method of integrating factors. The integrating factor, denoted by μ(t), is calculated as:
μ(t) = e^(∫P(t) dt)
In our case, P(t) = -1, so the integrating factor is:
μ(t) = e^(∫(-1) dt)
μ(t) = e^(-t)
We multiply the entire differential equation by the integrating factor e^(-t):
e^(-t) * (dy/dt) + e^(-t) * (-y) = e^(-t) * (-3A * e^(-2t))
e^(-t) * (dy/dt) - e^(-t) * y = -3A * e^(-3t)
The left-hand side of the equation is now the derivative of the product of y and the integrating factor, i.e., d/dt(y * e^(-t)). Thus, we can rewrite the equation as:
d/dt(y * e^(-t)) = -3A * e^(-3t)
To solve for y(t), we integrate both sides of the equation with respect to t:
∫d/dt(y * e^(-t)) dt = ∫(-3A * e^(-3t)) dt
y * e^(-t) = A * e^(-3t) + C₂
Here, C₂ is the constant of integration. Finally, we multiply both sides of the equation by e^(t) to isolate y(t):
y(t) = e^(t) * (A * e^(-3t) + C₂)
y(t) = A * e^(-2t) + C₂ * e^(t)
This is the general solution for y(t). It consists of two terms: the first term, A * e^(-2t), is a particular solution that depends on the solution of x(t), and the second term, C₂ * e^(t), is a homogeneous solution. The constant C₂ determines the behavior of y(t) as t increases.
Having solved both differential equations, we can now present the general solution for the system. The solutions for x(t) and y(t) are:
x(t) = A * e^(-2t)
y(t) = A * e^(-2t) + C₂ * e^(t)
Where A and C₂ are arbitrary constants. These constants can be determined if initial conditions are provided, such as the values of x and y at a specific time t. The solution for x(t) shows an exponential decay, while the solution for y(t) is a combination of an exponential decay term and an exponential growth term. The behavior of the system depends on the values of the constants A and C₂. The general solution represents a family of solutions, each corresponding to a different pair of values for A and C₂. To find a unique solution, we need additional information in the form of initial conditions.
To determine a particular solution for the system, we need initial conditions. Initial conditions provide specific values for the functions x(t) and y(t) at a particular time, usually t = 0. For example, suppose we have the initial conditions:
x(0) = x₀
y(0) = y₀
Where x₀ and y₀ are given constants. We can use these conditions to find the values of the arbitrary constants A and C₂ in the general solutions. First, we apply the initial condition x(0) = x₀ to the solution for x(t):
x(0) = A * e^(-2 * 0)
x₀ = A * e^(0)
x₀ = A
Thus, we find that A = x₀. Next, we apply the initial condition y(0) = y₀ to the solution for y(t):
y(0) = A * e^(-2 * 0) + C₂ * e^(0)
y₀ = A * e^(0) + C₂ * e^(0)
y₀ = A + C₂
Substituting A = x₀, we get:
y₀ = x₀ + C₂
C₂ = y₀ - x₀
Now we have found the values of both constants A and C₂ in terms of the initial conditions x₀ and y₀. Substituting these values back into the general solutions, we obtain the particular solutions:
x(t) = x₀ * e^(-2t)
y(t) = x₀ * e^(-2t) + (y₀ - x₀) * e^(t)
These particular solutions satisfy both the differential equations and the given initial conditions. They provide a unique description of the system's behavior given the starting values x₀ and y₀. Understanding how initial conditions influence the solution is crucial in many applications, as it allows us to predict the system's future state based on its initial state.
In this article, we have demonstrated a systematic approach to solve a system of differential equations. The system under consideration was:
dx/dt = -2x
dy/dt = -3x + y
We began by solving the first equation, a separable differential equation, using the method of separation of variables. This yielded the general solution x(t) = A * e^(-2t), where A is an arbitrary constant. We then substituted this solution into the second equation, transforming it into a first-order linear differential equation. This equation was solved using the method of integrating factors, resulting in the general solution y(t) = A * e^(-2t) + C₂ * e^(t), where C₂ is another arbitrary constant. The general solution for the system is thus:
x(t) = A * e^(-2t)
y(t) = A * e^(-2t) + C₂ * e^(t)
We further discussed how initial conditions can be used to determine particular solutions. By providing the values of x(0) and y(0), we were able to find the specific values of the constants A and C₂, leading to unique solutions that satisfy both the differential equations and the initial conditions. Solving systems of differential equations is a fundamental skill in many areas of science and engineering. The methods presented in this article provide a solid foundation for tackling more complex systems. Understanding the behavior of solutions, such as exponential decay and growth, is crucial for modeling and predicting real-world phenomena. This step-by-step guide aimed to clarify the process and enhance comprehension of solving such systems.