Question

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

Short answer

Question: Find the function F(x) that satisfies the given third-order differential equation \(F^{\prime\prime\prime}(x) = 672x^{5} + 24x\), and the following initial conditions: \(F^{\prime\prime}(0)=0\), \(F^{\prime}(0)=2\), and \(F(0)=1\). Answer: The function F(x) that satisfies the given differential equation and initial conditions is \(F(x) = 2x^8 + x^4 + 2x + 1\).

Step-by-step solution

Integrate the differential equation

We have the third derivative given as: \(F^{\prime\prime\prime}(x) = 672x^{5} + 24x\). Integrate with respect to \(x\) to find the second derivative: $$F^{\prime\prime}(x) = \int (672x^5 + 24x) dx = \frac{672}{6}x^{6} + \frac{24}{2}x^2 + C_1$$ Simplifying, we get: $$F^{\prime\prime}(x) = 112x^6 + 12x^2 + C_1$$

Use the initial condition \(F^{\prime\prime}(0)=0\) to find \(C_1\)

Use the initial condition: \(F^{\prime\prime}(0) = 112(0)^6 + 12(0)^2 + C_1 = 0\). This gives us \(C_1 = 0\). So we have: $$F^{\prime\prime}(x) = 112x^6 + 12x^2$$

Integrate again to find the first derivative

Now we integrate \(F^{\prime\prime}(x)\) with respect to \(x\) to find \(F^{\prime}(x)\): $$F^{\prime}(x) = \int (112x^6 + 12x^2) dx = \frac{112}{7}x^{7} + \frac{12}{3}x^3 + C_2$$ Simplifying, we get: $$F^{\prime}(x) = 16x^7 + 4x^3 + C_2$$

Use the initial condition \(F^{\prime}(0)=2\) to find \(C_2\)

Use the initial condition: \(F^{\prime}(0) = 16(0)^7 + 4(0)^3 + C_2 = 2\). This gives us \(C_2 = 2\). So we have: $$F^{\prime}(x) = 16x^7 + 4x^3 + 2$$

Integrate once more to find \(F(x)\)

Finally, we integrate \(F^{\prime}(x)\) with respect to \(x\) to find \(F(x)\): $$F(x) = \int (16x^7 + 4x^3 + 2) dx = \frac{16}{8}x^{8} + \frac{4}{4}x^4 + 2x + C_3$$ Simplifying, we get: $$F(x) = 2x^8 + x^4 + 2x + C_3$$

Use the initial condition \(F(0)=1\) to find \(C_3\)

Use the initial condition: \(F(0) = 2(0)^8 + (0)^4 + 2(0) + C_3 = 1\). This gives us \(C_3 = 1\). So we have: $$F(x) = 2x^8 + x^4 + 2x + 1$$

Write the final solution

The function F that satisfies the given differential equation and initial conditions is: $$F(x) = 2x^8 + x^4 + 2x + 1$$

Key concepts

Integration

When faced with differential equations, integration is a crucial technique used to reverse differentiation. This process is essential for finding functions at lower derivative levels. In our exercise, we start with the third derivative, which means we need to integrate three times to uncover the original function. By integrating the third derivative \(F^{\prime\prime\prime}(x) = 672x^5 + 24x\), we move down to the second derivative. The general formula for integration involves finding the antiderivative of each term:

• For \(672x^5\), the antiderivative is \(\frac{672}{6}x^6\).
• For \(24x\), the antiderivative is \(\frac{24}{2}x^2\).

By computing these, you determine the expression for \(F^{\prime\prime}(x)\). Each integration step often adds a constant (like \(C_1\), \(C_2\), etc.), which is determined using initial conditions.

Initial Conditions

Initial conditions are specific values given for the function or its derivatives at certain points, which help find the constants that appear during integration. These conditions are vital for reaching a specific solution that fits a real-world scenario. In the given problem:

• \(F^{\prime\prime}(0) = 0\) is used to find \(C_1\).
• \(F^{\prime}(0) = 2\) is used to find \(C_2\).
• \(F(0) = 1\) is used to find \(C_3\).

Each of these values allows us to plug into the respective equations obtained after each integration step to solve for the constants \(C_1\), \(C_2\), and \(C_3\). This step ensures that the function \(F(x)\) accurately matches the initial conditions and solves the problem completely.

Third Derivative

The third derivative of a function, denoted as \(F^{\prime\prime\prime}(x)\), provides information on the rate of change of the rate of change of the rate of change of the original function. In simpler terms, it is the derivative of the second derivative, which itself is the derivative of the first derivative. For polynomial functions, the third derivative tends to reduce the degree of terms considerably compared to the original function.
In our problem, we were given the third derivative of the function as \(672x^5 + 24x\). This form indicates the underlying function \(F(x)\) has polynomial terms. Integrating this expression helps us backtrack to find the second derivative, then the first, and finally the original function \(F(x)\). This derivative is essential, as starting from a higher derivative like the third ensures all lower derivative forms need to be computed to get the actual function.

Solving Differential Equations

Solving differential equations involves finding a function based on its derivative(s) and any initial or boundary conditions provided. The task often includes several integrations and uses initial conditions to determine any constants arisen during the process.For the problem presented, solving involves:

• Identifying given differential equations and initial conditions.
• Integrating to obtain lower derivatives step-by-step.
• Applying initial conditions to solve for constants.
• Compiling the function with all known information.

Through systematic integration and careful usage of the given initial conditions, the solution \(F(x) = 2x^8 + x^4 + 2x + 1\) is found. Each step is vital for building a function that meets all conditions and correctly represents the phenomenon described by the differential equation.