Question

determine a suitable form for Y(\(t\)) if the method of undetermined coefficients is to be used. Do not evaluate the constants. \(y^{\prime \prime \prime}-2 y^{\prime \prime}+y^{\prime}=t^{3}+2 e^{t}\)

Short answer

Answer: The suitable form for Y(t) is \(Y(t) = At^{3} + Bt^{2} + Ct + D + Ee^{t}\), where A, B, C, D, and E are constants to be determined.

Step-by-step solution

Identify the terms in the driving (nonhomogeneous) term

In the given differential equation, the nonhomogeneous term is \(t^{3}+2 e^{t}\). We can see that it has two different types of terms: a polynomial term \(t^{3}\) and an exponential term \(2 e^{t}\). We will consider both of these terms separately when determining the form of Y(\(t\)).

Determine the form for the polynomial term

The polynomial term in the driving term is \(t^{3}\). The form for Y(\(t\)) will have the same degree of the polynomial, which is 3. Therefore, the form for Y(\(t\)) corresponding to \(t^{3}\) is \(At^{3} + Bt^{2} + Ct + D\), where A, B, C, and D are constants to be determined.

Determine the form for the exponential term

The exponential term in the driving term is \(2 e^{t}\). The form for Y(\(t\)) will have the same base of the exponential function, which is \(e^{t}\). However, we need to make sure that it is linearly independent from the polynomial terms that we have already found in step 2. Therefore, the form for Y(\(t\)) corresponding to \(2 e^{t}\) is \(Ee^{t}\), where E is another constant to be determined.

Step 4:Combine the forms for polynomial and exponential terms

Now that we have the individual forms for both the polynomial term and the exponential term, we can combine them to obtain the suitable form for Y(\(t\)). So, the form for Y(\(t\)) is: \(Y(t) = At^{3} + Bt^{2} + Ct + D + Ee^{t}\) We do not need to evaluate the constants(A, B, C, D, and E) as per the exercise requirements. The form of Y(\(t\)) has been successfully found.

Key concepts

Nonhomogeneous Differential Equation

A nonhomogeneous differential equation is a type of differential equation that includes a non-zero term on its right-hand side. This non-zero term is referred to as the nonhomogeneous or driving term, which distinguishes it from a homogeneous differential equation, where the right-hand side is zero. In our exercise, the equation presented is:\[ y^{\prime \prime \prime} - 2y^{\prime \prime} + y^{\prime} = t^{3} + 2e^{t} \]The right-hand side, \(t^{3} + 2e^{t}\), is the nonhomogeneous part. This term includes different types of functions, specifically polynomial and exponential components, necessitating particular attention in solving the equation with undetermined coefficients. Identifying the nature of the nonhomogeneous term is crucial for the accurate determination of the form of the particular solution \(Y(t)\).

Polynomial Solution

When dealing with a polynomial solution within the context of nonhomogeneous differential equations, we are often required to match the degree of the polynomial in the nonhomogeneous term. This helps us hypothesize part of the form of the particular solution.In our exercise, the nonhomogeneous term contains a polynomial part \(t^3\). This tells us that our particular solution \(Y(t)\) needs to have a counterpart of at least the same degree. Thus, the polynomial solution part of \(Y(t)\) is expressed as:- \[ At^3 + Bt^2 + Ct + D \]where \(A, B, C,\) and \(D\) are constants that will be determined once we further analyze the solution. The reason we include terms down to the constant \(D\) is to potentially capture any linear or constant solutions needed to satisfy other parts of the equation.

Exponential Solution

The exponential solution is another piece we must consider due to the presence of an exponential term in the nonhomogeneous part of our differential equation. When we have an exponential function such as \(2e^t\), it suggests that our particular solution \(Y(t)\) should include an exponential component.For the given \(2e^t\), we propose the form:- \[ Ee^t \]Here, \(E\) is a constant that will need to be determined through subsequent calculations. The goal is to ensure that this form is linearly independent from any other parts of the solution, which often means checking against terms derived under the polynomial portion. This independence ensures we can solve for the constants without conflicts or redundancy.

Linear Independence

Linear independence is a concept in differential equations that ensures the components of our proposed solution do not repeat or overlap in a manner that prevents us from solving for unique coefficients. For solutions to work properly, all proposed terms in \(Y(t)\) must remain linearly independent.When combined, all polynomial and exponential parts of \(Y(t)\) have distinguished levels of independence, as the polynomial terms \(At^3 + Bt^2 + Ct + D\) differ structurally from the exponential \(Ee^t\). This separation allows us to avoid situations like two copies of the same term appearing, which would otherwise make solving for unique constants impossible.In this exercise, ensuring that \(Ee^t\) remains independent of polynomial terms is straightforward, given their inherently different mathematical structures. This careful consideration is crucial in reaching a complete and accurate solution for \(Y(t)\).