Question

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{m}(x)=672 x^{5}+24 x ; F^{\prime \prime}(0)=0, F^{\prime}(0)=2, F(0)=1$$

Short answer

Given the differential equation and initial conditions, we have determined that the function satisfying these conditions is: $$F(x) = 2x^8 + x^4 + 2x + 1$$

Step-by-step solution

Integrate the m-th derivative once

Integrate the given m-th derivative \(F^m(x) = 672x^5 + 24x\) with respect to x: $$F^{m-1}(x) = \int{672x^5}dx + \int{24x}dx = 112x^6 + 12x^2 + C_1$$ At this point, we do not yet know what \(m-1\) is, so we need to find this before we can proceed.

Determine the value of \(m-1\)

From the initial condition, we know that \(F^{\prime\prime}(0) = 0\). This tells us that the second derivative has no constant term as that term in any higher-order derivatives will not disappear after integration. Thus, \(m-1\) must be 2, which means that \(m\) is 3. In other words, the given formula is the third derivative of function \(F(x)\). So now, we have $$F^{\prime\prime}(x) = 112x^6 + 12x^2 + C_1$$

Integrate the second derivative

Integrate the second derivative with respect to x to find the first derivative \(F^{\prime}(x)\): $$F^{\prime}(x) = \int{(112x^6 + 12x^2 + C_1)}dx = 16x^7 + 4x^3 + C_1x + C_2$$

Apply the initial condition for \(F^{\prime}(0)\)

We are told that \(F^{\prime}(0) = 2\). We can substitute this into the equation we found for \(F^{\prime}(x)\) to determine the constant \(C_2\): $$2 = 16(0)^7 + 4(0)^3 + C_1(0) + C_2 \Rightarrow C_2 = 2$$ Now we know that: $$F^{\prime}(x) = 16x^7 + 4x^3 + C_1x + 2$$

Integrate the first derivative

Integrate the first derivative with respect to x to find the function \(F(x)\): $$F(x) = \int{(16x^7 + 4x^3 + C_1x + 2)}dx = 2x^8 + x^4 + \frac{1}{2}C_1x^2 + 2x + C_3$$

Apply the initial condition for \(F(0)\)

Use \(F(0) = 1\) to determine the constant \(C_3\): $$1 = 2(0)^8 + (0)^4 + \frac{1}{2}C_1(0)^2 + 2(0) + C_3 \Rightarrow C_3 = 1$$ Now we have: $$F(x) = 2x^8 + x^4 + \frac{1}{2}C_1x^2 + 2x + 1$$

Find \(C_1\) from the second initial condition

We know that \(F^{\prime\prime}(0) = 0\), and we have the expression for \(F^{\prime\prime}(x)\). So, we can use this condition to find \(C_1\): $$0 = 112(0)^6 + 12(0)^2 + C_1$$ Since all terms with x become zero: $$C_1 = 0$$ Finally, we found our function \(F(x)\): $$F(x) = 2x^8 + x^4 + 2x + 1$$

Key concepts

Initial Conditions

In solving differential equations, initial conditions play an essential role in determining the specific solution we are seeking. Initial conditions are the prescribed values for the function or its derivatives at a particular point.
These conditions help us identify the arbitrary constants produced during the integration process. Without initial conditions, we can only determine the general form of the solution rather than a specific one.
When working with the exercise, we see the use of initial conditions like:

• \(F''(0) = 0\)
• \(F'(0) = 2\)
• \(F(0) = 1\)

These initial values are vital for solving the differential equation as they help identify unknown constants such as \(C_1\), \(C_2\), and \(C_3\). Without these values given at \(x = 0\), the final function \(F(x) = 2x^8 + x^4 + 2x + 1\) could not be precisely found.

Integration

Integration is a fundamental concept in calculus used to find functions when given their derivatives. In the context of differential equations, it involves reversing the differentiation process.
This means you need to find a function whose derivative matches the given function or expression.
In this exercise, the solution involves multiple integrations since we are dealing with higher-order derivatives as the equation is given as \(F^{m}(x) = 672x^5 + 24x\).
The process involves these steps:

• Integrate the given m-th derivative to find the (m-1)th derivative.
• Repeat the integration process with obtained derivatives until reaching the actual function.

Each integration step adds an unknown constant \(C\), which is later solved using initial conditions. Integration helps convert a derivative back into its original function form, making it an essential tool in solving differential equations.

Higher-order Derivatives

In calculus, higher-order derivatives represent successive applications of differentiation to a function. The n-th order derivative provides information about the rates of change beyond just the slope, such as concavity and inflection points.
This exercise requires working backwards from the third derivative to find the function \(F(x)\).
Here, we know that the third derivative \(F^{m}(x) = 672x^5 + 24x\) is provided. From this, you will:

• Integrate to retrieve the second derivative.
• Apply initial conditions to resolve constants and further reduce the derivative.
• Continue this process until returning to the original function.

In this task, the ability to reason about higher-order derivatives helps us progressively understand how each layer of differentiation reveals a larger aspect of the function. The task of backtracking from higher derivatives through integration is a common strategy to solve complex differential equations.