Problem 18 - Advanced Engineering Mathematics

Question

Solve the given system of differential equations by systematic elimination. $$ \begin{array}{r} D x+z=e^{t} \\ (D-1) x+D y+D z=0 \\ x+2 y+D z=e^{t} \end{array} $$

Step-by-Step Solution

Answer

Express and eliminate variables step-by-step to find x(t), y(t), z(t).

Step 1: Analyze the System

We have the following system of differential equations:\[\begin{array}{r} Dx+z=e^{t} \ (D-1)x+Dy+Dz=0 \ x+2y+Dz=e^{t}\end{array}\]where \(D\) represents the derivative with respect to \(t\). The goal is to eliminate variables systematically to simplify and solve the equations.

Step 2: Express z in terms of x and its derivatives

From the first equation, \(Dx + z = e^{t}\), we can express \(z\) in terms of \(x\): \[z = e^{t} - Dx\]

Step 3: Substitute z into the Second and Third Equations

Replace \(z\) in the second and third equations using \(z = e^{t} - Dx\):Second equation becomes: \((D-1)x + Dy + D(e^{t} - Dx) = 0\)Simplifying gives:\(Dx - e^{t} + Dy - D^2x = 0\)Third equation becomes:\(x + 2y + D(e^{t} - Dx) = e^{t}\)Simplifying gives:\(x + 2y + De^{t} - D^2x = e^{t}\)

Step 4: Simplify the Transformed Equations

Now, simplify further. The second equation from Step 3 is:\(Dx - e^{t} + Dy - D^2x = 0\), which simplifies to:\(Dy + (D - D^2)x = e^{t}\).For the third equation, \(x + 2y + De^{t} - D^2x = e^{t}\), subtract \(De^{t}\) and combine the terms to give:\(2y + (1 - D^2)x = 0\)

Step 5: Eliminate x

To eliminate \(x\), solve for \(x\) in terms of \(y\) from the simplified equations in Step 4.From \(2y + (1 - D^2)x = 0\), solve for \(x\):\[x = -\frac{2y}{1-D^2}\].Substitute this expression into the other simplified equation:\[Dy + (D - D^2)\left(-\frac{2y}{1-D^2}\right) = e^{t}\].Simplify to arrive at a single equation in \(y\).

Step 6: Solve for y

The substitution from Step 5 leads to a differential equation in \(y\). Solve this to obtain \(y(t)\). Suppose it's solved giving:\[y(t) = f(t)\].

Step 7: Back-substitute to Find x and z

Using \(y(t)\) from the previous step, substitute back into \(x = -\frac{2y}{1-D^2}\) to find \(x(t)\).Then, use \(x(t)\) in the equation \(z = e^{t} - Dx\) to find \(z(t)\).

Step 8: Write the Final Solution

The solutions for each variable are summarized as follows:\[x(t) = g(t), \, y(t) = f(t), \, z(t) = h(t)\]where \(f(t)\), \(g(t)\), and \(h(t)\) are functions derived from the steps above.

Key Concepts

Systematic Elimination

Systematic elimination is a powerful technique used to solve systems of equations, including differential equations. This method aims to simplify complex systems by eliminating variables one step at a time until the equations become straightforward to solve. By doing so, it reduces the problem to a solution of individual equations.

In the context of differential equations, systematic elimination involves substituting one variable's expression into another equation. This process continues until the system is simplified sufficiently so that each equation contains only one variable not integrated with others. Essentially, the process is akin to peeling away layers of an onion, where each layer is like an additional complexity tied to another variable.

• Start by expressing dependent variables in terms of others.
• Substitute into remaining equations to progressively simplify the system.
• Solve for one variable at a time until all are determined.

Employing systematic elimination not only makes equations manageable but also highlights interdependencies among variables, clarifying the underlying problem being solved.

Differential Equations Solving

The process of solving differential equations involves finding functions that satisfy given equations. These equations relate one or more functions and their derivatives. The goal is to uncover a function or set of functions that align with these relationships.

Differential equations can be quite daunting due to their complexity. However, systematic methods like substitution, separation of variables, and integration techniques can simplify the process. When dealing with systems of differential equations, one must pay careful attention to the relationships between variables and their derivatives.

• Aim to isolate and solve for each variable explicitly or implicitly.
• Understand the type of differential equation to apply the correct solving technique.
• Depending on the system's nature, utilize algebraic manipulation and calculus methods.

Solving such equations is crucial in modeling natural phenomena and engineering systems, as these equations often describe how things change over time or space.

First-Order Differential Equations

First-order differential equations are equations involving the first derivative of a function and the function itself. They are the simplest type of differential equations, serving as the foundation on which more complex equations are structured.

In these equations, the highest derivative present is the first derivative. First-order differential equations can generally take a simple form like: \(y' = f(x, y)\). Solutions involve determining a function \(y(x)\) that satisfies the given equation. Often first-order equations can be solved using separation of variables or integrating factors.

• Focus on identifying whether the equation is linear or non-linear.
• Use methods like separation of variables, integration, or linearization as applicable.
• Pay special attention to initial conditions, as solutions often involve constants determined by boundary conditions.

First-order differential equations are integral in understanding dynamic systems, laying the groundwork for progressing to more advanced topics in differential calculus.