Question

For which \(k\) does the system of equations \(\dot{x}_{1}=x_{1}, \dot{x}_{2}=k x_{2}\) on the whole plane have a non-constant first integral?

Short answer

Answer: The system has a non-constant first integral if \(k \neq 1\).

Step-by-step solution

Rewrite the system in terms of the first derivatives

First, let's rewrite the system in terms of the first derivatives \(\dot{x}_1\) and \(\dot{x}_2\): $$\frac{d}{dt}x_1 = x_1$$ $$\frac{d}{dt}x_2 = kx_2$$

Solve the first equation

Solve the first equation for \(x_1\): $$\frac{d}{dt}x_1 = x_1$$ This is a separable equation, which can be solved by dividing both sides by \(x_1\) and integrating: $$\int \frac{1}{x_1} dx_1 = \int dt$$ Applying the integration yields: $$\ln |x_1| = t + C_1$$ Where \(C_1\) is an arbitrary constant. Exponentiate both sides to find the solution for \(x_1\): $$x_1(t) = e^{t+C_1} = Ae^t$$ Where \(A=e^{C_1}\) is another arbitrary constant.

Solve the second equation

Now, solve the second equation for \(x_2\): $$\frac{d}{dt}x_2 = kx_2$$ This equation is also separable, and we can solve it similarly: $$\int \frac{1}{x_2} dx_2 = \int k dt$$ The integrations yield: $$\ln |x_2| = kt + C_2$$ Where \(C_2\) is an arbitrary constant. Exponentiate both sides to find the solution for \(x_2\): $$x_2(t) = e^{kt+C_2} = Be^{kt}$$ Where \(B=e^{C_2}\) is another arbitrary constant.

Find the non-constant first integral

A first integral is a function \(I(x_1, x_2)\) that remains constant along any trajectory of the system. We can find it by eliminating the time variable \(t\) from the solutions for \(x_1\) and \(x_2\): The simplest non-constant function is their quotient: $$I(x_1, x_2) = \frac{x_1}{x_2} = \frac{Ae^t}{Be^{kt}} = \frac{AB^k e^{-kt}}{B}$$ Thus, for a non-constant first integral to exist, the exponent of \(e\) must not be zero. In other words, the system will have a non-constant first integral if \(k \neq 1\).

State the answer

The system of equations has a non-constant first integral if \(k \neq 1\).

Key concepts

Separable Differential Equations

Separable differential equations are a special class of ordinary differential equations where you can isolate the variables on either side of the equation. In other words, they can be written in the form \(\frac{dy}{dx} = g(x)h(y)\). Here, the function \(g(x)\) involves only the independent variable, x, and the function \(h(y)\) involves only the dependent variable, y.

To solve such equations, one must 'separate' the variables, which typically involves dividing both sides by \(h(y)\) and multiplying by \(dx\), effectively organizing the equation so that each variable and its differential are on opposite sides. After separating the variables, integrate both sides: \(int \frac{1}{h(y)} dy = \int g(x) dx\). You'll end up with two antiderivatives, often involving a constant of integration.

When we apply this method to \(dx_1/dt = x_1\) and \(dx_2/dt = kx_2\), you're performing a classic variable separation to solve these first-order linear differential equations. The beauty of this method lies in its simplicity, which makes it a powerful tool for students trying to understand not just the steps, but also the concepts behind solving differential equations.

Exponential Growth

Exponential growth occurs when the rate of growth is proportional to the quantity present. In other words, the bigger something is, the faster it grows. This property is characteristic of systems where the function representing the quantity doubles at regular intervals, forming a distinctive exponential curve when plotted over time.

In our exercise, \(x_1(t) = Ae^t\) and \(x_2(t) = Be^{kt}\) are both examples of exponential functions, which solve the corresponding differential equations. \(e\) refers to Euler's number, an important mathematical constant approximately equal to 2.71828. Here, \(t\) represents time, and \(A\) and \(B\) are constants related to the initial conditions of each function at \(t = 0\).

The presence of \(k\) in the second solution modifies the growth rate of \(x_2\). If \(k>1\), \(x_2\) grows faster than \(x_1\), and if \(0

Solving Differential Equations

Solving differential equations is a cornerstone of understanding dynamic processes in engineering, physics, biology, economics, and many other fields. Differential equations relate a function with its derivatives, representing how a particular quantity changes over time or space.

The general idea is to integrate both sides of the equation, often involving initial conditions to solve for any constants that appear from the integration process. The solutions to these equations provide a formula that describes the behavior of the system modeled by the equation.

In the given exercise, solving the system involves integrating the differential equations for \(x_1\) and \(x_2\). The solutions for these variables are functions of time, which are then used to determine the behavior of the system's trajectories. Because of their properties and solutions, differential equations can model an incredibly wide range of phenomena — from simple processes like cooling coffee to complex systems like weather patterns or the stability of ecosystems. Mastering the art of solving such equations is crucial for students, as it opens doors to understanding the mathematical framework that underpins much of the world around us.

Solving Differential Equations

Solving differential equations is a cornerstone of understanding dynamic processes in engineering, physics, biology, economics, and many other fields. Differential equations relate a function with its derivatives, representing how a particular quantity changes over time or space.

The general idea is to integrate both sides of the equation, often involving initial conditions to solve for any constants that appear from the integration process. The solutions to these equations provide a formula that describes the behavior of the system modeled by the equation.

In the given exercise, solving the system involves integrating the differential equations for \(x_1\) and \(x_2\). The solutions for these variables are functions of time, which are then used to determine the behavior of the system's trajectories. Because of their properties and solutions, differential equations can model an incredibly wide range of phenomena — from simple processes like cooling coffee to complex systems like weather patterns or the stability of ecosystems. Mastering the art of solving such equations is crucial for students, as it opens doors to understanding the mathematical framework that underpins much of the world around us.