Solving $y' = -4\sqrt{x} + 24e^{-6x} - 9\sin X$ With Initial Condition (0, 8)

Introduction

In this article, we will explore the process of solving the differential equation $y' = -4\sqrt{x} + 24e^{-6x} - 9\sin x$, given that the solution passes through the point $(0, 8)$. This type of problem falls under the category of first-order differential equations, specifically those that can be solved by direct integration. We will walk through the steps of finding the general solution by integrating the given equation and then applying the initial condition to determine the particular solution. Understanding how to solve differential equations is crucial in many fields of science and engineering, as they are used to model various phenomena such as population growth, radioactive decay, and the motion of objects. This problem provides a good example of how to apply integration techniques and initial conditions to find a unique solution.

The differential equation presented, $y' = -4\sqrt{x} + 24e^{-6x} - 9\sin x$, is a blend of power, exponential, and trigonometric functions. Solving it requires a firm grasp of integration rules and techniques associated with these individual functions. Furthermore, the initial condition, a pivotal piece of information, guides us from the general solution to the particular solution, one that precisely fits the given point (0, 8). This article will elucidate each step, providing clarity on the application of integration and the incorporation of the initial condition to arrive at the final solution. We will also delve into the significance of initial conditions in determining unique solutions to differential equations, and how they translate the family of possible solutions into a single, definitive one.

Finding the General Solution

The first step in solving the given differential equation is to find the general solution. Since $y'$ represents the derivative of $y$ with respect to $x$, we can find $y$ by integrating both sides of the equation with respect to $x$. This involves applying the fundamental theorem of calculus, which states that integration is the reverse process of differentiation. The general solution will contain an arbitrary constant of integration, denoted as $C$, which accounts for the family of possible solutions. This constant will be determined later using the initial condition provided.

To find the general solution of the differential equation $y' = -4\sqrt{x} + 24e^{-6x} - 9\sin x$, we integrate each term separately. The integral of $-4\sqrt{x}$ is found by rewriting $\sqrt{x}$ as $x^{\frac{1}{2}}$ and applying the power rule for integration, which states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$. Thus, the integral of $-4x^{\frac{1}{2}}$ becomes $-4 \cdot \frac{x^{\frac{3}{2}}}{\frac{3}{2}} = -\frac{8}{3}x^{\frac{3}{2}}$. Next, we integrate $24e^{-6x}$. The integral of $e^{kx}$ is $\frac{1}{k}e^{kx}$, so the integral of $24e^{-6x}$ is $24 \cdot \frac{e^{-6x}}{-6} = -4e^{-6x}$. Finally, we integrate $-9\sin x$. The integral of $\sin x$ is $-\cos x$, so the integral of $-9\sin x$ is $-9(-\cos x) = 9\cos x$. Combining these results, we obtain the general solution:

$y = -\frac{8}{3}x^{\frac{3}{2}} - 4e^{-6x} + 9\cos x + C$

This general solution represents a family of curves, each differing by the value of the constant $C$. To find the specific solution that satisfies the given condition, we need to determine the value of $C$. This is where the initial condition comes into play, allowing us to pinpoint the unique solution that passes through the specified point. The process of finding this particular solution is the next critical step in solving the differential equation.

Applying the Initial Condition

To find the particular solution, we need to use the given initial condition, which is the point $(0, 8)$. This means that when $x = 0$, $y = 8$. We substitute these values into the general solution we found in the previous section to solve for the constant of integration, $C$. The initial condition acts as a constraint that selects one specific solution from the infinite family of solutions represented by the general solution. This process is essential for obtaining a unique solution that accurately describes the specific situation being modeled by the differential equation.

Substituting $x = 0$ and $y = 8$ into the general solution $y = -\frac{8}{3}x^{\frac{3}{2}} - 4e^{-6x} + 9\cos x + C$, we get:

$8 = -\frac{8}{3}(0)^{\frac{3}{2}} - 4e^{-6(0)} + 9\cos(0) + C$

Simplifying this equation, we have:

$8 = 0 - 4e^{0} + 9(1) + C$

Since $e^0 = 1$ and $\cos(0) = 1$, the equation becomes:

$8 = -4 + 9 + C$

$8 = 5 + C$

Solving for $C$, we find:

$C = 8 - 5 = 3$

Thus, the constant of integration is $C = 3$. This value of $C$ is crucial as it defines the specific solution that passes through the point $(0, 8)$. By substituting this value back into the general solution, we obtain the particular solution that satisfies both the differential equation and the initial condition. This particular solution represents the unique curve that fits the given requirements, providing a complete and accurate solution to the problem.

Determining the Particular Solution

Now that we have found the constant of integration, $C = 3$, we can substitute this value back into the general solution to obtain the particular solution. This particular solution is the unique solution that satisfies both the differential equation and the initial condition. It represents a specific curve in the family of solutions, distinguished by the fact that it passes through the point $(0, 8)$. The process of finding this particular solution highlights the importance of initial conditions in determining a unique and meaningful solution to a differential equation.

Substituting $C = 3$ into the general solution $y = -\frac{8}{3}x^{\frac{3}{2}} - 4e^{-6x} + 9\cos x + C$, we get the particular solution:

$y = -\frac{8}{3}x^{\frac{3}{2}} - 4e^{-6x} + 9\cos x + 3$

This equation represents the solution to the initial value problem (IVP). The IVP consists of the differential equation and the initial condition, and its solution is the particular solution we have just found. This solution is a function that, when differentiated, yields the original differential equation, and it also satisfies the condition that $y(0) = 8$. This ensures that the solution is both mathematically correct and relevant to the specific context of the problem. The particular solution provides a complete and unambiguous answer to the problem, making it a crucial step in solving differential equations.

Conclusion

In conclusion, we have successfully solved the differential equation $y' = -4\sqrt{x} + 24e^{-6x} - 9\sin x$, given the initial condition that the solution passes through the point $(0, 8)$. We began by finding the general solution through integration, which involved applying the power rule, the exponential integral, and the trigonometric integral. This general solution included an arbitrary constant of integration, $C$, representing a family of possible solutions. Then, we applied the initial condition to determine the specific value of $C$, which allowed us to find the particular solution.

The particular solution we found is $y = -\frac{8}{3}x^{\frac{3}{2}} - 4e^{-6x} + 9\cos x + 3$. This solution satisfies both the differential equation and the initial condition, making it the unique solution to the initial value problem. The process of solving this problem demonstrates the importance of both integration techniques and the application of initial conditions in finding solutions to differential equations. Understanding these concepts is fundamental in various fields, as differential equations are widely used to model real-world phenomena.

This exercise illustrates the power of differential equations in describing and predicting the behavior of systems. The initial condition played a crucial role in narrowing down the infinite possibilities presented by the general solution to a single, definitive solution. This solution accurately represents the function that not only satisfies the given rate of change but also passes through the specified point, providing a comprehensive understanding of the system's behavior. The ability to solve differential equations is a valuable skill in many scientific and engineering disciplines, enabling the analysis and modeling of dynamic systems with precision and accuracy.

Final Answer

The final answer is:

B. $y=-\frac{8}{3} x^{\frac{3}{2}}-4 e^{-6 x}+9 \cos x+3$