Problem 4 - Advanced Engineering Mathematics

Question

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}-4 y=1 \\ &\frac{d y}{d t}+x=2 \end{aligned} $$

Step-by-Step Solution

Answer

Solve two related variables using differential substitutions and simplification.

Step 1: Express One Variable in Terms of the Other

From the first equation \( \frac{d x}{d t} - 4y = 1 \), solve for \( \frac{d x}{d t} \):\[ \frac{d x}{d t} = 4y + 1. \]

Step 2: Substitute in the Second Equation

Substitute \( x \) from the second equation \( \frac{d y}{d t} + x = 2 \) to express \( x \) in terms of \( \frac{d y}{d t} \): \( x = 2 - \frac{d y}{d t} \).

Step 3: Combine Equations for One Variable

Using these expressions, substitute \( x = 2 - \frac{d y}{d t} \) into the equation found in Step 2:\[ \frac{d x}{d t} = 4y + 1 \Rightarrow 4y + 1 = 2 - \frac{d y}{d t}. \] Solve this to find\[ \frac{d y}{d t} = x - 2. \] Now combine:\[ \frac{d x}{d t} = 4y + 1 = 4 \left( 2 - \frac{d y}{d t} \right) + 1. \] Simplifying leads to:\[ \frac{d x}{d t} = 8 - 4\frac{d y}{d t} + 1. \]

Step 4: Eliminate \( y \) and Solve

Substitute \( \frac{d y}{d t} \) from the second equation into the first to obtain:\[ 5. \Rightarrow \frac{d x}{d t} = 9 - 4(2 - \frac{d y}{d t}). \] Substitute \( \frac{d x}{d t} = 4 \frac{d y}{d t} + 9 \) into:\( x = 2 - \frac{d y}{d t} \).

Step 5: Solve for x and y

Solve the combined equation:\[ x = 2 - \frac{d y}{d t} \qquad and \qquad \frac{d x}{d t} = 4 \frac{d y}{d t} + 1. \] Simplify to find values.

Key Concepts

Systematic Elimination

Systematic elimination is a powerful technique for solving systems of equations, including systems of differential equations. The goal is to eliminate one of the variables to simplify the system into a single equation. This method makes it easier to find solutions for complex systems where variables are interdependent. Let's look at how this works for the given exercise.

• First, express one of the original equations in terms of one of the variables, such as expressing \( \frac{dx}{dt} \) from the first equation.
• Next, substitute this expression into the other equation, effectively reducing two equations to one by eliminating the chosen variable.
• Solve the resulting single equation to find the value of one of the variables, which can then be substituted back to find the other variable.

Using systematic elimination, both \(x\) and \(y\) can be found, simplifying the process considerably.

First-Order Differential Equations

First-order differential equations involve derivatives of the first degree and are the simplest form of differential equations to solve. They often represent processes where changes accumulate over time, such as speed or growth. Here, both equations given in the system are first-order differential equations.

• The first equation, \( \frac{dx}{dt} - 4y = 1 \), describes a rate of change in terms of \(y\).
• The second equation, \( \frac{dy}{dt} + x = 2 \), represents another rate of change, this time involving \(x\).

Understanding first-order differential equations is key to solving the system because each variable's behavior and interdependency are expressed through these equations. By isolating and integrating these equations, solutions can describe the system's dynamics over time.

Substitution Method

The substitution method is a specific approach used in conjunction with systematic elimination to solve systems of equations. With this method, you replace one variable with another expression to simplify the problem.

• First, identify an equation where one variable can be expressed in terms of the other. In the given system, \(x\) can be expressed as \(x = 2 - \frac{dy}{dt}\).
• Second, substitute this expression into the other equation. This simplifies the system by removing one of the variables, leaving a single equation involving only one variable.

Once you solve the single equation for the remaining variable, you can substitute back to find the first one. This method is deeply connected with systematic elimination, as both work towards simplifying the system to make finding solutions straightforward.