Chapter 3: Problem 18
For the given differential equation, $$ y^{\prime \prime}-y^{\prime}=t^{2}\left(2+e^{t}\right) $$
Short Answer
Expert verified
The general solution of the given differential equation is:
$$
y(t) = (C_1 + C_2e^{t}) + (t^2 + t^3 + 4te^t)
$$
where \(C_1\) and \(C_2\) are arbitrary constants.
Step by step solution
01
Find the complementary function
Start by solving the homogeneous equation:
$$
y^{\prime \prime}-y^{\prime}=0
$$
In order to solve, we'll assume a solution of the form:
$$
y = e^{rt}
$$
Differentiate twice to get:
$$
y^{\prime} = re^{rt} \\
y^{\prime \prime} = r^2e^{rt}
$$
Now plug these expressions into the homogeneous equation:
$$
r^2e^{rt} - re^{rt} = 0
$$
Simply the equation by factoring:
$$
e^{rt}(r^2 - r) = 0
$$
This equation is satisfied for:
$$
e^{rt} \neq 0 \Rightarrow r^2 - r = 0 \Rightarrow r(r-1) = 0
$$
So the possible values of r are 0 and 1. The complementary function \(y_c\) is the linear combination of these solutions:
$$
y_c = C_1e^{0t} + C_2e^{t} = C_1 + C_2e^{t}
$$
02
Find a particular integral
Find a particular solution of the given non-homogeneous differential equation. We'll use a trial function based on the forcing function:
$$
f(t) = t^2(2+e^t)
$$
The trial function has the form:
$$
y_p = At^2 + Bt^3 + Cte^t
$$
Now, find the first and second derivatives of \(y_p\):
$$
y_p^{\prime} = 2At + 3Bt^2 + Ct(1+t)e^t \\
y_p^{\prime \prime} = 2A + 6Bt + C(1+2t)e^t
$$
Plug these expressions into the original non-homogeneous differential equation:
$$
(2A + 6Bt + C(1+2t)e^t) - (2At + 3Bt^2 + Ct(1+t)e^t) = t^2(2+e^t)
$$
Upon expanding and comparing coefficients of \(t^n e^t\), we find three equations:
$$
-2A + 2B = 0 \Rightarrow B = A \\
6B - 3B + C = 2 \\
C - 3B = 1
$$
Solve the system of equations to find:
$$
A = 1, B = 1, C = 4
$$
Construct the particular integral \(y_p\) using these coefficients:
$$
y_p = t^2 + t^3 + 4te^t
$$
03
Combine the complementary function and particular integral to get the general solution
Combine \(y_c\) and \(y_p\) to obtain the general solution:
$$
y(t) = y_c + y_p = (C_1 + C_2e^{t}) + (t^2 + t^3 + 4te^t)
$$
This is the general solution to the given non-homogeneous second-order linear differential equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complementary Function
When dealing with differential equations, the complementary function is a key component in finding the solution to homogeneous equations. Here we have a homogeneous equation like \(y'' - y' = 0\). The idea is to find solutions that naturally satisfy this without any external influence—basically the part of the system's response that exists without any forcing functions.
We begin by proposing a solution of the form \(y = e^{rt}\). This is a common method used for such differential equations because exponentials make it easy to handle derivatives. After substituting this trial solution into the equation, we end up with a characteristic equation \(r(r-1) = 0\). Solving this yields roots \(r = 0\) and \(r = 1\).
Each root gives us a basic solution: \(e^{0t} = 1\) and \(e^{t}\). These solutions reflect the inherent behavior of the system without any non-homogeneous term acting on it. The complementary function, therefore, is a linear combination of these basic solutions: \(y_c = C_1 + C_2e^t\). Here, \(C_1\) and \(C_2\) are arbitrary constants determined by initial or boundary conditions.
We begin by proposing a solution of the form \(y = e^{rt}\). This is a common method used for such differential equations because exponentials make it easy to handle derivatives. After substituting this trial solution into the equation, we end up with a characteristic equation \(r(r-1) = 0\). Solving this yields roots \(r = 0\) and \(r = 1\).
Each root gives us a basic solution: \(e^{0t} = 1\) and \(e^{t}\). These solutions reflect the inherent behavior of the system without any non-homogeneous term acting on it. The complementary function, therefore, is a linear combination of these basic solutions: \(y_c = C_1 + C_2e^t\). Here, \(C_1\) and \(C_2\) are arbitrary constants determined by initial or boundary conditions.
Non-Homogeneous Equation
A non-homogeneous equation differs from a homogeneous one in that it includes a non-zero term which represents an external condition or force acting on the system. The equation in our example is \(y'' - y' = t^2(2+e^t)\). The term \(t^2(2+e^t)\) is what makes the equation non-homogeneous and represents an external factor influencing the system's behavior.
Understanding non-homogeneous equations is essential because they model real-world systems where outside forces act upon variables. These could range from external electrics in circuits to external forces in mechanical systems. The solution process involves solving both the homogeneous part, via the complementary function, and the particular solution which accounts for the specific external terms. Combining these gives the full response of the system, reflecting both its natural tendencies and how it reacts to input or changes from the external environment.
This is also a standard approach in engineering and physics, where systems are rarely isolated and usually respond to a combination of inherent characteristics and external influences.
Understanding non-homogeneous equations is essential because they model real-world systems where outside forces act upon variables. These could range from external electrics in circuits to external forces in mechanical systems. The solution process involves solving both the homogeneous part, via the complementary function, and the particular solution which accounts for the specific external terms. Combining these gives the full response of the system, reflecting both its natural tendencies and how it reacts to input or changes from the external environment.
This is also a standard approach in engineering and physics, where systems are rarely isolated and usually respond to a combination of inherent characteristics and external influences.
Particular Solution
Finding a particular solution to a non-homogeneous differential equation requires us to address the external force or term in the equation. For the equation, \(y'' - y' = t^2(2+e^t)\), the particular solution, \(y_p\), is specifically crafted to account for the \(t^2(2+e^t)\) part.
We use a method known as "undetermined coefficients," where we propose a trial function of a form similar to the non-homogeneous term. In this example, a reasonable guess is \(y_p = At^2 + Bt^3 + Cte^t\) as it matches the structure of \(t^2(2+e^t)\).
Differentiating this form provides us with expressions for \(y_p'\) and \(y_p''\). Substituting these into the original equation helps identify coefficients \(A\), \(B\), and \(C\), ensuring \(y_p\) satisfies the equation. After some calculations, we determine that \(A = 1\), \(B = 1\), and \(C = 4\). Hence, our particular solution is \(y_p = t^2 + t^3 + 4te^t\). This process of selecting and refining the trial function is essential for isolating how a system's response is shaped by external influences.
We use a method known as "undetermined coefficients," where we propose a trial function of a form similar to the non-homogeneous term. In this example, a reasonable guess is \(y_p = At^2 + Bt^3 + Cte^t\) as it matches the structure of \(t^2(2+e^t)\).
Differentiating this form provides us with expressions for \(y_p'\) and \(y_p''\). Substituting these into the original equation helps identify coefficients \(A\), \(B\), and \(C\), ensuring \(y_p\) satisfies the equation. After some calculations, we determine that \(A = 1\), \(B = 1\), and \(C = 4\). Hence, our particular solution is \(y_p = t^2 + t^3 + 4te^t\). This process of selecting and refining the trial function is essential for isolating how a system's response is shaped by external influences.
General Solution
The general solution of a differential equation is the combination of the complementary function and the particular solution. It represents the total behavior of the system under both its own inherent tendencies and outside influences.
For our example, the general solution \(y(t)\) combines the complementary function \(y_c = C_1 + C_2e^t\) with the particular solution \(y_p = t^2 + t^3 + 4te^t\). Thus, the complete solution is:
\[y(t) = (C_1 + C_2e^t) + (t^2 + t^3 + 4te^t)\]
This expression captures all the system behaviors. The complementary function covers how the system would evolve naturally without external influences, while the particular solution reflects the system's response to the given non-homogeneous part, \(t^2(2+e^t)\).
The constants \(C_1\) and \(C_2\) are typically determined from initial or boundary conditions, making the general solution a crucial result that can be applied to specific problems or real-world scenarios, tailoring the solution precisely to the situation.
For our example, the general solution \(y(t)\) combines the complementary function \(y_c = C_1 + C_2e^t\) with the particular solution \(y_p = t^2 + t^3 + 4te^t\). Thus, the complete solution is:
\[y(t) = (C_1 + C_2e^t) + (t^2 + t^3 + 4te^t)\]
This expression captures all the system behaviors. The complementary function covers how the system would evolve naturally without external influences, while the particular solution reflects the system's response to the given non-homogeneous part, \(t^2(2+e^t)\).
The constants \(C_1\) and \(C_2\) are typically determined from initial or boundary conditions, making the general solution a crucial result that can be applied to specific problems or real-world scenarios, tailoring the solution precisely to the situation.
