91Ó°ÊÓ

A non-homogeneous differential equation includes a term that is not solely dependent on the function and its derivatives. In our problem, the differential equation is given by: \[ y''' + y' = e^t. \]Notice the presence of \(e^t\) which makes this equation non-homogeneous. Here's why understanding this is important:

• Handles real-world scenarios: Many physical phenomena can't be explained by homogeneous equations because there's always some external influence or input, represented by the non-homogeneous term.
• Diverse solutions: Includes both a complementary solution and a particular solution to cover various influences.

The key to solving such equations is tackling two parts: the complementary solution (treating it as homogeneous) and the particular solution (factoring in the non-homogeneous term).

The particular solution, \(y_p(t)\), is a specific solution that satisfies the entire non-homogeneous differential equation. For our equation, \( y''' + y' = e^t \), the non-homogeneous term is \( e^t \). To find the particular solution, guess a form that mimics the non-homogeneous term.

Since the non-homogeneous term is \( e^t \), we try a solution form: \[ y_p(t) = Ae^t. \]Taking derivatives, we get:\[ y_p'(t) = Ae^t, \]\[ y_p'''(t) = Ae^t. \]Substitute these into the differential equation:\[ Ae^t + Ae^t = e^t, \]Simplifying, we find:\[ 2Ae^t = e^t, \]which yields:\[ A = \frac{1}{2}. \]Thus, the particular solution is:\[ y_p(t) = \frac{1}{2} e^t. \]This particular solution adds to our understanding of how the non-homogeneous term \(e^t\) influences the system.

The complementary solution, \(y_c(t)\), solves the associated homogeneous equation: \[ y''' + y' = 0. \]Assuming a solution of the form \( y = e^{rt} \) transforms the equation into a characteristic equation:\[ r^3 + r = 0. \]By factoring out \( r \), we get:\[ r(r^2 + 1) = 0. \]This gives the roots: \( r = 0, \, i, \, -i \).
Thus, the complementary solution is:\[ y_c(t) = C_1 + C_2 \, \text{cos}(t) + C_3 \, \text{sin}(t). \]Here's why the complementary solution is vital:

• Forms base: It provides the fundamental solutions under no external influence.
• Completeness: Combined with the particular solution, it covers all scenarios.

This complementary solution reflects the natural response of the system to initial disturbances.

Initial conditions define the specific behavior of the differential equation's solution at the beginning. In our equation, we have:\[ y(0) = y'(0) = y''(0) = 0. \]To incorporate these, substitute the initial conditions into the general solution. Here's the general solution formed by adding both the complementary and particular solutions:\[ y(t) = C_1 + C_2 \, \text{cos}(t) + C_3 \, \text{sin}(t) + \frac{1}{2} e^t. \]Using the initial conditions:\[ y(0) = 0 \Rightarrow C_1 + C_2 + \frac{1}{2} = 0 \Rightarrow C_1 + C_2 = -\frac{1}{2}, \]\[ y'(0) = 0 \Rightarrow C_3 + \frac{1}{2} = 0 \Rightarrow C_3 = -\frac{1}{2}, \]\[ y''(0) = 0 \Rightarrow -C_2 + \frac{1}{2} = 0 \Rightarrow C_2 = \frac{1}{2}. \]Solving these, we get:\[ C_1 = -1. \]So, our final solution respecting initial conditions is:\[ y(t) = -1 + \frac{1}{2} \, \text{cos}(t) - \frac{1}{2} \, \text{sin}(t) + \frac{1}{2} e^t. \]Initial conditions are critical to tailoring the general solution to the specifics you'd encounter in real-world situations.

In our differential equation, the coefficients of y and its derivatives are constants, as seen in \[ y''' + y' = e^t. \]Having constant coefficients simplifies the process:

• Predictable solutions: Making it easier to apply standard methods like characteristic equations.
• Ease of factoring: Facilitates the derivation of characteristic equations.

In our equation, because the coefficients are constants, we easily formed the characteristic equation: \[ r^3 + r = 0. \]By solving this, we obtained the roots necessary for our complementary solution. This predictability simplifies the entire solution process, allowing you to apply systematic methods to obtain precise results. This consistency helps stabilize calculations, making these types of differential equations more manageable than those with variable coefficients.