Question

For each differential equation, (a) Find the complementary solution. (b) Formulate the appropriate form for the particular solution suggested by the method of undetermined coefficients. You need not evaluate the undetermined coefficients. $$y^{(4)}+4y=e^t\sin t$$

Short answer

The complementary solution is: $y_c(t)=c_1{e}^{r_{1}t}+c_2{e}^{r_{2}t}+c_3{e}^{r_{3}t}+c_4{e}^{r_{4}t}$, where $r_1=4^{\frac{1}{4}}{e}^{i\frac{\pi }{4}}$, $r_2=4^{\frac{1}{4}}{e}^{i\frac{3\pi }{4}}$, $r_3=4^{\frac{1}{4}}{e}^{i\frac{5\pi }{4}}$, and $r_4=4^{\frac{1}{4}}{e}^{i\frac{7\pi }{4}}$. The appropriate form for the particular solution is: $y_p(t)=e^t(A\cos t+B\sin t)$, where $A$ and $B$ are coefficients to be determined.

Step-by-step solution

Find the complementary solution

To find the complementary solution, we first need to find the characteristic equation for the given differential equation: $$r^4+4=0$$ To solve this equation, let's express it as: $$r^4=-4$$ We can rewrite the equation in terms of complex numbers: $$r^4=4e^{i\pi +2\pi ki}$$ where $k=0,1,2,3$ and $i$ is the imaginary unit. Now, let's take the fourth root of both sides ($r=e^{(1/4)\ln (4e^{i\pi +2\pi ki})}$): $$r=4^{\frac{1}{4}}{e}^{i(\frac{\pi }{4}+\frac{\pi k}{2})}$$ This gives us the four distinct roots: $$r_1=4^{\frac{1}{4}}{e}^{i\frac{\pi }{4}},r_2=4^{\frac{1}{4}}{e}^{i\frac{3\pi }{4}},r_3=4^{\frac{1}{4}}{e}^{i\frac{5\pi }{4}},r_4=4^{\frac{1}{4}}{e}^{i\frac{7\pi }{4}}$$ Then, the complementary solution is: $$y_c(t)=c_1{e}^{r_{1}t}+c_2{e}^{r_{2}t}+c_3{e}^{r_{3}t}+c_4{e}^{r_{4}t}$$ where $c_1$, $c_2$, $c_3$, and $c_4$ are constants.

Find the appropriate form for the particular solution

Now, let's look at the forcing function: $e^t\sin t$. We want to find a form for the particular solution that does not overlap with the complementary solution. In this case, we can choose the following form: $$y_p(t)=e^t(A\cos t+B\sin t)$$ where $A$ and $B$ are coefficients to be determined (undetermined coefficients). Note that we do not need to find the values of $A$ and $B$ in this exercise. The given appropriate form for the particular solution is already in the correct form for the method of undetermined coefficients. In conclusion, the complementary solution is $y_c(t)=c_1{e}^{r_{1}t}+c_2{e}^{r_{2}t}+c_3{e}^{r_{3}t}+c_4{e}^{r_{4}t}$ and the appropriate form for the particular solution is $y_p(t)=e^t(A\cos t+B\sin t)$.

Key concepts

Complementary Solution

When solving a differential equation, it's essential to understand the concept of the complementary solution. The complementary solution, often denoted as $y_c(t)$, is derived from the homogeneous part of the differential equation, which does not take into consideration any external forces or inputs.
In technical terms, it is the general solution to the homogeneous differential equation associated with the original non-homogeneous equation. The process to find the complementary solution is straightforward: you find the characteristic equation of the homogeneous part, solve it for its roots, and based on these roots, the complementary solution is constructed.
In the provided example, the characteristic equation $r^4+4=0$ leads to complex roots due to the equation's structure. These roots are typically expressed in exponential terms using Euler's formula, combining real and imaginary components. The complementary solution is then a combination of terms of the form $c_k{e}^{r_{k}t}$, where $c_k$ represents constants and $r_k$ the roots. Each term in the complementary solution corresponds to a root of the characteristic equation, and together, they encompass all the solution behaviors without external influences.

Particular Solution

Opposite to the complementary solution is the particular solution, denoted as $y_p(t)$, which addresses the specific external input or forcing function in the differential equation. The particular solution only concerns the non-homogeneous portion of the equation, incorporating the 'particular' effects of the given forcing term.
To find a suitable form for the particular solution, we observe the nature of the forcing function. For the forcing function $e^t\sin t$, we predict a solution form that mirrors this behavior but avoids duplicating any component of the complementary solution. After deciding on the form that typically includes undetermined coefficients, we substitute this assumed form back into the differential equation to solve for these coefficients. Although, in the current task, we only need to suggest the form and not actually determine these coefficients.
The specific choice of $y_p(t)=e^t(A\cos t+B\sin t)$ is significant because it captures the essential characteristics of the forcing function while complying with the method of undetermined coefficients. However, we should always check the proposed form against the complementary solution to ensure no overlap, which could invalidate the method.

Method of Undetermined Coefficients

The method of undetermined coefficients is an ingenious approach used for finding the particular solution of linear non-homogeneous differential equations with constant coefficients. It is applicable when the forcing function is a simple function like polynomials, exponentials, sines, and cosines, or sums and products thereof.
To apply this method effectively, one generally follows a prescribed set of forms for the assumed particular solution, which corresponds to the type of the forcing function. For our given problem, we assume $y_p(t)$ has the form $e^t(A\cos t+B\sin t)$ because the forcing function is $e^t\sin t$. You then substitute this assumed form into the differential equation, leading to equations for the undetermined coefficients $A$ and $B$. The goal is to choose a form that makes it possible to solve for these coefficients through algebraic manipulation.
The choice of form is critical: it must respect the original equation's structure and ensure the solution is linearly independent from the complementary solution. If the initial guess for the particular solution has a term that is similar to any part of the complementary solution, we need to modify the assumed form by multiplying by the independent variable until we achieve independence. This process ensures the validity of the superposition principle, allowing us to combine the complementary and particular solutions to form the general solution.

Characteristic Equation

The characteristic equation is a cornerstone in solving linear differential equations with constant coefficients. It is a polynomial equation whose solutions, or roots, lay the groundwork for constructing the complementary solution.
Derived from the differential equation by replacing each derivative of the function by a power of an unknown $r$, the characteristic equation encapsulates the behavior of the homogenous part of the system. For instance, the characteristic equation $r^4+4=0$ from the original exercise converts the fourth-order differential equation into a simple polynomial that we can solve for $r$.
Solving these characteristic equations often involves complex numbers, as they might have imaginary roots. In our example, solving for $r$ results in complex roots that, when plugged back into the general form of the solution, yield sinusoidal functions indicative of oscillatory motion, exponential growth, or decay. The formation of the characteristic equation and the subsequent finding of its roots are vital steps in constructing the full solution to the differential equation – comprising both the complementary and the particular solutions.