The given differential equation is:

$y''' + y'' + y' + y = e^{-t} + 4t.$

This is a non-homogeneous linear differential equation of third order. The general solution can be found in two main steps:

1. Solve the homogeneous equation:

$y''' + y'' + y' + y = 0.$

2. Find a particular solution to the non-homogeneous equation:

$y''' + y'' + y' + y = e^{-t} + 4t.$

Step 1: Solving the homogeneous equation

The characteristic equation for the homogeneous equation is obtained by assuming a solution of the form $y = e^{rt}$, leading to:

$r^3 + r^2 + r + 1 = 0.$

Factoring this cubic polynomial, we look for roots of the characteristic equation. Using synthetic division or trial and error, $r = -1$ is a root. Now, factor $r + 1$ out:

$r^3 + r^2 + r + 1 = (r + 1)(r^2 + 1).$

This gives the roots:

• $r = -1$ (real root),
• $r = \pm i$ (complex roots).

The general solution to the homogeneous equation is: $y_h(t) = c_1 e^{-t} + c_2 \cos(t) + c_3 \sin(t).$

Step 2: Solving the non-homogeneous equation

We need to find a particular solution $y_p(t)$ to:

$y''' + y'' + y' + y = e^{-t} + 4t.$

We handle each part of the right-hand side separately:

• For the $e^{-t}$ term, we already have $e^{-t}$ as a solution to the homogeneous equation, so we multiply by $t$ to form a particular solution of the form $At e^{-t}$.
• For the $4t$ term, we try a solution of the form $Bt^2 + Ct$.

Thus, the particular solution is:

$y_p(t) = At e^{-t} + Bt^2 + Ct.$

Substitute this into the original equation and solve for $A$, $B$, and $C$ by equating the coefficients of like terms.

Step 3: General Solution

The general solution is the sum of the homogeneous and particular solutions:

$y(t) = y_h(t) + y_p(t) = c_1 e^{-t} + c_2 \cos(t) + c_3 \sin(t) + At e^{-t} + Bt^2 + Ct.$

Would you like to go through the detailed calculation for finding the constants $A$, $B$, and $C$?

Related questions:

1. What is the method of undetermined coefficients for solving non-homogeneous differential equations?
2. How do we factor higher-order polynomials like $r^3 + r^2 + r + 1$?
3. What is the significance of complex roots in the characteristic equation of a differential equation?
4. Can variation of parameters be used to solve this equation, and how does it compare to the method of undetermined coefficients?
5. How do we handle non-homogeneous terms like polynomials or exponentials in differential equations?

Tip:

When solving non-homogeneous differential equations, always solve the homogeneous equation first to ensure you can appropriately modify the form of the particular solution if necessary.