Problem 3 - Advanced Engineering Mathematics

Question

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=-y+t \\ &\frac{d y}{d t}=x-t \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system is \( x(t) = c_1 e^t + c_2 e^{-t} + 2 \) and \( y(t) = -c_1 e^t + c_2 e^{-t} + t \).

Step 1: Express Derivatives

From the system of differential equations, we have two equations:1. \( \frac{d x}{d t} = -y + t \)2. \( \frac{d y}{d t} = x - t \).Our goal is to express \( y \) and \( x \) in terms of their respective derivatives.

Step 2: Substitute and Simplify

Substitute the expression for \( y \) from the first equation into the second.- From \( \frac{d x}{d t} = -y + t \), we get \( y = -\frac{d x}{d t} + t \).- Substitute in \( \frac{d y}{d t} = x - t \): \[ \frac{d y}{d t} = x - t = \frac{d^2 x}{d t^2}, \] as \( y = -\frac{d x}{d t} + t \).

Step 3: Solve the Second Order Equation

Now, we solve the second order differential equation:\[\frac{d^2 x}{d t^2} = x - 2t.\]The complementary function is obtained from \( \frac{d^2 x}{d t^2} = x \), which yields a characteristic equation with roots \( \lambda^2 - 1 = 0 \). This gives a solution of:\[x_h(t) = c_1 e^t + c_2 e^{-t}.\]For the particular integral of \( x - 2t \), assume \( x_p = A + Bt \). Substituting gives \( A = 2 \) and \( B = 0 \). Thus,\[x_p(t) = 2.\]The general solution is:\[x(t) = c_1 e^t + c_2 e^{-t} + 2.\]

Step 4: Find y(t) in Terms of x(t)

Using the expression for \( y \):\[y = -\frac{d x}{d t} + t,\]we substitute \( x(t) \) into this equation.The derivative \( \frac{d x}{d t} = c_1 e^t - c_2 e^{-t} \).Substitute into the equation for \( y \):\[y(t) = -c_1 e^t + c_2 e^{-t} + t.\]

Step 5: Final Solutions

The solutions for the system of differential equations are:\( x(t) = c_1 e^t + c_2 e^{-t} + 2 \) and \( y(t) = -c_1 e^t + c_2 e^{-t} + t \).These solutions describe the behavior of \( x \) and \( y \) over time, depending on the values of the constants \( c_1 \) and \( c_2 \), determined by initial conditions.

Key Concepts

Systematic Elimination

Systematic elimination is a technique to simplify and solve systems of differential equations.
This process involves strategically substituting variables to reduce the system to a simpler form. Typically, the goal is to first express each variable in terms of derivatives, then systematically substitute one equation into another.

• Identify the relationships between the variables in the given equations.
• Substitute variables to eliminate one of them, often transforming the system into a single differential equation.
• Solve the resulting simplified equation.

This approach can transform a coupled system into separate equations, which are often easier to handle individually. It's crucial in making complex systems more manageable by reducing dimensions.

Second Order Differential Equations

Solving a system often involves reducing it to a second order differential equation. Such equations involve second derivatives, like \(\frac{d^2 x}{d t^2}\). These can represent phenomena like acceleration in physical contexts.
When you reduce a system to the form \(\frac{d^2 x}{d t^2} = x - 2t\), a clear method emerges:

• The homogeneous part, \(\frac{d^2 x}{d t^2} = x\), represents the system's natural behavior without external influence.
• The particular part, given non-homogeneous terms like \(-2t\), shows how the system responds to external inputs.

Understanding these components helps in constructing solutions piece-by-piece, capturing both inherent dynamics and specific external effects.

Characteristic Equation

The characteristic equation is essential for finding the "homogeneous" solution part of a differential equation. It comes from assuming solutions of the form \(e^{\lambda t}\), leading directly to characteristic equations like \(\lambda^2 - 1 = 0\). Here’s how it works:

• The characteristic equation results from substituting trial solutions into the differential equation.
• Solving it yields roots that determine the form of the homogeneous solution.
• For example, \(\lambda^2 - 1 = 0\) gives roots \(\lambda = 1\) and \(-1\), leading to solutions \(c_1 e^t + c_2 e^{-t}\).

This step is vital as it captures how solutions behave due to the system's natural properties, without additional influences.

General Solution

The general solution of a differential equation combines both the homogeneous solution and a particular solution. This captures the complete behavior of the system over time.
For instance, in our example,

• The homogeneous solution \(x_h(t) = c_1 e^t + c_2 e^{-t}\) reflects natural dynamics.
• A particular solution takes into account constant external forces, like \(x_p(t) = 2\).

Thus, the general solution \(x(t) = c_1 e^t + c_2 e^{-t} + 2\) represents all possible scenarios depending on initial conditions (given by \(c_1\) and \(c_2\)). Such combinations ensure that solutions are both flexible and complete, accounting for every possible behavior of the system.