91Ó°ÊÓ

First-order differential equations are mathematical expressions involving derivatives of a function with respect to one variable. They are termed 'first-order' because they involve the first derivative of the function. In this exercise, we looked at two equations with derivatives indicating rates of change: the rate at which \(y_1\) changes with respect to \(t\) and the rate at which \(y_2\) changes with respect to \(x\). These equations were presented:

• \( y_1' = 4y_2 + 3e^{3t} \)
• \( y_2' = 2y_2 - 15e^{-3x} \)

Such equations are fundamental in describing linear systems and various natural phenomena, like population growth or radioactive decay. The challenge lies in determining the function from its rate of change. This often involves techniques such as the integrating factor method, exemplified in this problem.

The integrating factor method is a powerful technique for solving first-order linear non-homogeneous differential equations. The idea is to multiply the entire differential equation by a carefully chosen function, called the integrating factor, which simplifies the equation into something more straightforward to solve.

For the equation \( y_2' = 2y_2 - 15e^{-3x} \), we selected \( e^{2x} \) as the integrating factor. This factor is computed from the coefficient of \( y_2 \) in the equation. By multiplying through by the integrating factor, we can convert the left-hand side into the derivative of a product:

• \( e^{2x}y_2' - 2e^{2x}y_2 = -15e^{-x+2x} \)
• Which simplifies to \( \frac{d}{dx}(e^{2x}y_2) = -15e^{x} \)

This transformation results in an equation that can be integrated directly, allowing us to solve for \( y_2 \). This process highlights the elegance and utility of the integrating factor in solving such equations.

Linear non-homogeneous differential equations contain terms that are not dependent on the function or its derivatives alone; they also have a `non-homogeneous' term or input. These equations often arise in practical contexts where an external force or input is affecting the system.

In our work, \( y_2' = 2y_2 - 15e^{-3x} \) is a great example of such an equation. The term \(-15e^{-3x}\) represents an external input affecting the variable \( y_2 \). Solving these equations requires addressing both the homogeneous part (which would result if the non-homogeneous term were zero) and the particular part, which is due to the non-homogeneous term.

The general solution combines both elements:

• A homogeneous solution: \( Ce^{-2x} \)
• A particular solution: \(-15e^{-x} \)

Hence, the complete solution is \( y_2 = -15e^{-x} + 17e^{-2x} \), fully accounting for every aspect of the differential equation.

Initial conditions are specific values provided for the function at a certain point, crucial for identifying the constants within the general solution of differential equations. They "anchor" the function at particular points, ensuring a unique solution appropriate to the context described.

In the given problem, initial conditions are:

• \( y_1(0) = 2 \)
• \( y_2(0) = 2 \)

Applying these conditions allowed us to solve for the constants within our solutions for \( y_1 \) and \( y_2 \). For instance, in solving for \( y_2 \), plugging the initial value into the general equation gave us \( C = 17 \), tailoring the solution exactly to
\( y_2 = -15e^{-x} + 17e^{-2x} \).

Likewise for \( y_1 \), substituting its condition solved for another constant, leading to a specific solution fitting all conditions faithfully. Initial conditions ensure the solutions are meaningful and precisely adapted to the problem described.