Question

Solve the differential equation \(y^{\prime \prime \prime}=2\) by computing successive antiderivatives. What is the order of this differential equation? How many arbitrary constants arise in the antidifferentiation solution process?

Short answer

Answer: The order of the given differential equation is 3, and there are 3 arbitrary constants in the solution.

Step-by-step solution

Identify the given differential equation

The given differential equation is: \(y^{\prime \prime \prime} = 2\) It is a third-order equation, as the highest derivative is the third derivative of y.

Compute the first antiderivative

To find the first antiderivative, integrate both sides of the equation with respect to x: \(\int y^{\prime \prime \prime} dx = \int 2 dx\) As a result, we get: \(y^{\prime \prime} = 2x + C_1\)

Compute the second antiderivative

Next, we will find the second antiderivative by integrating both sides of the above equation with respect to x: \(\int y^{\prime \prime} dx = \int (2x + C_1) dx\) This gives us: \(y^{\prime} = x^2 + C_1x + C_2\)

Compute the third antiderivative

Now, we will find the third antiderivative by integrating both sides of the equation with respect to x: \(\int y^{\prime} dx = \int (x^2 + C_1x + C_2) dx\) As a result, we get the solution to the differential equation: \(y = \frac{1}{3}x^3 + \frac{1}{2}C_1x^2 + C_2x + C_3\)

Determine the order and the number of arbitrary constants

Since the given differential equation is third-order, it is of order 3. In the process of finding the solution, we have encountered three arbitrary constants, \(C_1\), \(C_2\), and \(C_3\). Thus, there are three arbitrary constants in the solution. Final Answer: The order of the differential equation is 3, and there are 3 arbitrary constants in the solution: \(C_1\), \(C_2\), and \(C_3\). The general solution is given by: \(y = \frac{1}{3}x^3 + \frac{1}{2}C_1x^2 + C_2x + C_3\).

Key concepts

Antiderivatives

Antiderivatives are functions that reverse the process of differentiation. If you have a function and know its derivative, an antiderivative is the original function before it was differentiated.
For example, if the derivative of a function is a constant 2, a corresponding antiderivative would be a function whose derivative is 2, such as 2x.
In the context of differential equations, computing antiderivatives helps us uncover the original functions.
Each time we compute an antiderivative, we include an unknown constant, known as an arbitrary constant, because many different functions can share the same derivative.

• The antiderivative of a constant 2 is 2x + C, where C is the arbitrary constant.
• Similarly, for a polynomial function like 2x, the antiderivative would be x² + C.

Thus, finding antiderivatives involves integrating the function, which is the opposite of taking the derivative.

Order of Differential Equation

The order of a differential equation is determined by the highest derivative present in the equation. It's an important characteristic as it tells us how many times we need to differentiate an unknown function to arrive at the given equation.
In our exercise, the differential equation is \(y''' = 2\). Here, \(y'''\) represents the third derivative of the function \(y\) with respect to \(x\).
Hence, it is a third-order differential equation because the highest order of derivative is three.

• An equation containing \(y''\) would be a second-order differential equation.
• An equation with only \(y'\) would be a first-order differential equation.

The order determines how many times one needs to compute antiderivatives to find the original function.

Arbitrary Constants

Arbitrary constants appear as integration constants when computing successive antiderivatives. They are "arbitrary" because their value isn't fixed by the integration process. Instead, they reflect the infinite number of possible functions that could satisfy a given equation when the derivative is known.
In the process of solving a differential equation through integration, each antiderivative introduces a new arbitrary constant.
For example:

• The first antiderivative of \(y''' = 2\) introduces \(C_1\).
• The second antiderivative introduces \(C_2\).
• The third antiderivative introduces \(C_3\).

Thus, for a third-order differential equation, we generally expect three arbitrary constants in the solution. These constants can be determined if specific conditions or values are known, which uniquely define the solution.

General Solution of Differential Equation

The general solution of a differential equation is an expression that includes arbitrary constants and captures all possible solutions of the equation. These arbitrary constants can be adjusted to fit specific initial or boundary conditions to find a particular solution.
In our exercise, the original differential equation \(y''' = 2\) has a general solution given by: \[y = \frac{1}{3}x^3 + \frac{1}{2}C_1x^2 + C_2x + C_3\]Here, \(C_1\), \(C_2\), and \(C_3\) are arbitrary constants.

• The term \(\frac{1}{3}x^3\) arises from integrating constant terms through each level of differentiation.
• The arbitrary constants allow the solution to represent a family of functions rather than a single function.

The general solution, therefore, provides the broadest form of the answer, encompassing all specific solutions that might be derived through further conditions.