Question

(a) Find the general solution of the differential equation. (b) Impose the initial conditions to obtain the unique solution of the initial value problem. (c) Describe the behavior of the solution $y(t)$ as $t\to -\mathrm{\infty }$ and as $t\to \mathrm{\infty }$. Does $y(t)$ approach $-\mathrm{\infty },+\mathrm{\infty }$, or a finite limit? $$y^{\prime \prime }-4y^{\prime }+3y=0,y(0)=-1,y^{\prime }(0)=1$$

Short answer

Answer: The unique solution of the initial value problem is $y(t)=e^{3t}-2e^t$. As $t\to -\mathrm{\infty }$, the solution $y(t)$ approaches 0, and as $t\to \mathrm{\infty }$, the solution $y(t)$ approaches $+\mathrm{\infty }$.

Step-by-step solution

Solve the characteristic equation

The given differential equation is: $$y^{\prime \prime }-4y^{\prime }+3y=0$$ To find the general solution of this homogeneous differential equation, we solve the associated characteristic equation: $$r^2-4r+3=0$$ This equation can be factored as $(r-3)(r-1)=0$, which gives us two distinct real roots, $r_1=3$ and $r_2=1$. So, the general solution of the given homogeneous differential equation is: $$y(t)=C_1{e}^{3t}+C_2{e}^t$$

Apply the initial conditions

To find the unique solution of the initial value problem, we need to apply the given initial conditions, $y(0)=-1$ and $y^{\prime }(0)=1$. Using the initial condition $y(0)=-1$, we find: $$y(0)=C_1{e}^{3\cdot 0}+C_2{e}^{1\cdot 0}=C_1+C_2$$ So, $C_1+C_2=-1$. Next, we compute the derivative of the general solution we found in step 1: $$y^{\prime }(t)=3C_1{e}^{3t}+C_2{e}^t$$ Now, using the initial condition $y^{\prime }(0)=1$, we find: $$y^{\prime }(0)=3C_1{e}^{3\cdot 0}+C_2{e}^{1\cdot 0}=3C_1+C_2$$ So, $3C_1+C_2=1$. Solving the system of linear equations $C_1+C_2=-1$ and $3C_1+C_2=1$, we find $C_1=1$ and $C_2=-2$. Thus, the unique solution of the initial value problem is: $$y(t)=e^{3t}-2e^t$$

Determine the behavior of the solution as $t\to -\mathrm{\infty }$ and $t\to \mathrm{\infty }$

To analyze the behavior of the solution as $t\to \pm \mathrm{\infty }$, we examine the terms $e^{3t}$ and $-2e^t$. As $t\to -\mathrm{\infty }$: - The term $e^{3t}$ approaches 0 because the exponent $3t$ approaches $-\mathrm{\infty }$. - The term $-2e^t$ also approaches 0 because the exponent $t$ approaches $-\mathrm{\infty }$. Hence, the limit of the solution as $t\to -\mathrm{\infty }$ is: $$\underset{t\to -\mathrm{\infty }}{lim}y(t)=\underset{t\to -\mathrm{\infty }}{lim}(e^{3t}-2e^t)=0$$ As $t\to \mathrm{\infty }$: - The term $e^{3t}$ approaches $+\mathrm{\infty }$ because the exponent $3t$ approaches $+\mathrm{\infty }$. - The term $-2e^t$ approaches $-\mathrm{\infty }$ because the exponent $t$ approaches $\mathrm{\infty }$, but with a smaller magnitude compared to the term $e^{3t}$. Hence, the limit of the solution as $t\to \mathrm{\infty }$ is: $$\underset{t\to \mathrm{\infty }}{lim}y(t)=\underset{t\to \mathrm{\infty }}{lim}(e^{3t}-2e^t)=+\mathrm{\infty }$$ In conclusion, as $t\to -\mathrm{\infty }$, the solution $y(t)$ approaches 0, and as $t\to \mathrm{\infty }$, the solution $y(t)$ approaches $+\mathrm{\infty }$.

Key concepts

Characteristic Equation

The characteristic equation is a foundation for understanding linear homogeneous differential equations. When we encounter a differential equation like $y^{″}-4y^{\prime }+3y=0$, we aim to express this second-order linear homogeneous differential equation in terms of its characteristic equation. This is achieved by assuming a solution of the form $y(t)=e^{rt}$, where $r$ is a value to be determined.

Plugging this assumption into the differential equation, we translate the problem into an algebraic equation, $r^2-4r+3=0$, by replacing $y^{″}$ with $r^2{e}^{rt}$, $y^{\prime }$ with $r{e}^{rt}$, and $y$ with $e^{rt}$. The roots of this characteristic equation, in this case, are real and distinct ($r_1=3$ and $r_2=1$), leading to the general solution involving exponentials of these roots. These steps crystallize the importance of the characteristic equation in revealing the behavior of the solution to the differential equation over time.

Initial Value Problem

An initial value problem involves finding a specific solution to a differential equation that not only satisfies the equation itself but also meets specified values at a given point in time, known as initial conditions.

In our example, the differential equation is accompanied by two such conditions: $y(0)=-1$ and $y^{\prime }(0)=1$. By applying these conditions to the general solution $y(t)=C_1{e}^{3t}+C_2{e}^t$, we can determine the coefficients $C_1$ and $C_2$, transforming the general solution into one that uniquely satisfies the given constraints. This unique solution deeply reflects the physical or theoretical context from which the differential equation originated and is tailored to the specified starting scenario.

Homogeneous Differential Equation

A homogeneous differential equation is one where all the terms are a function of the dependent variable and its derivatives, and it equals zero. The given equation, $y^{″}-4y^{\prime }+3y=0$, is a classic example of a second-order linear homogeneous differential equation.

These equations have the distinguished property that if you have two solutions, any linear combination of these solutions is also a solution – a principle known as the superposition principle. This concept ties back into our characteristic equation, as the general solution derived contains a linear combination of exponential functions associated with the roots of the characteristic equation. Hence, understanding homogeneous equations is critical for predicting system behavior in various scientific and engineering fields.

General Solution

The general solution to a differential equation encompasses all possible solutions, often containing arbitrary constants that are yet to be determined. In our tutorial exercise, the general solution is represented by $y(t)=C_1{e}^{3t}+C_2{e}^t$, wherein $C_1$ and $C_2$ are placeholders for any real numbers.

These constants are what permit the adaptation of the general solution to specific initial conditions, leading to a unique solution that perfectly fits those conditions. Understanding the general solution is thus vital; it's the broad answer from which we carve out the precise response we need with the use of initial conditions.

Limit Behavior

Limit behavior in the context of differential equations refers to the behavior of the solution as the independent variable, typically time (denoted as $t$), approaches either positive or negative infinity. This is an essential concept to discuss the long-term trends and stability of the solution.

For our example, we investigated what happens to $y(t)=e^{3t}-2e^t$ as $t$ trends towards negative and positive infinity. As $t$ approaches negative infinity, the exponential terms decay to zero, meaning the solution stabilizes to zero. Conversely, as $t$ goes to positive infinity, the dominant term $e^{3t}$ exponentially increases, indicating that the solution grows without bound. Such insights into the limit behavior are crucial for understanding the ultimate fate of the systems described by the differential equation, whether they are physical, biological, or financial in nature.