Finding Y In Terms Of X Solving Differential Equations
In the realm of calculus, one of the most fascinating and practical applications lies in solving differential equations. These equations, which relate a function to its derivatives, are the backbone of mathematical models that describe phenomena ranging from the motion of celestial bodies to the flow of fluids and the growth of populations. Among the various techniques for tackling differential equations, integration stands out as a fundamental tool. In this comprehensive exploration, we delve into the process of finding a function y in terms of x when we are given its derivative, dy/dx, and an initial condition. This problem exemplifies the power of integration in unraveling the relationship between a function and its rate of change, providing us with a complete picture of the function's behavior.
The Problem: A Step-by-Step Approach to Finding y(x)
Let's embark on our journey by stating the problem clearly. We are tasked with finding the function y in terms of x, given that the derivative of y with respect to x is expressed as dy/dx = 3x² - 6x + 2. Furthermore, we have an initial condition: when x = 0, the value of y is 7. This initial condition is crucial because it allows us to pinpoint a specific solution from the infinite family of functions that share the same derivative. Finding y in terms of x requires us to reverse the process of differentiation, which is precisely what integration accomplishes. We will begin by integrating both sides of the differential equation with respect to x. This step will introduce an arbitrary constant of integration, which we will then determine using the given initial condition. By substituting the values of x and y from the initial condition into the integrated equation, we can solve for the constant of integration. Finally, we will substitute the value of the constant back into the integrated equation, yielding the particular solution that satisfies both the differential equation and the initial condition. This step-by-step approach provides a clear roadmap for solving this type of problem, highlighting the interplay between differentiation, integration, and initial conditions.
Step 1: Integrating Both Sides of the Differential Equation
To find y in terms of x, we first need to reverse the process of differentiation. This is achieved through integration. We start with the given differential equation:
dy/dx = 3x² - 6x + 2
To isolate y, we integrate both sides of the equation with respect to x:
∫(dy/dx) dx = ∫(3x² - 6x + 2) dx
On the left-hand side, the integral of the derivative dy/dx with respect to x simply gives us y. On the right-hand side, we apply the power rule of integration, which states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration. This rule is fundamental to integrating polynomial terms. Applying the power rule to each term on the right-hand side, we get:
y = ∫3x² dx - ∫6x dx + ∫2 dx
y = 3∫x² dx - 6∫x dx + 2∫dx
y = 3(x³/3) - 6(x²/2) + 2x + C
y = x³ - 3x² + 2x + C
This result is a general solution to the differential equation. It represents a family of curves, each differing by the value of the constant of integration, C. To find the specific solution that satisfies our problem, we need to determine the value of C using the initial condition.
Step 2: Using the Initial Condition to Find the Constant of Integration
The initial condition provides us with a specific point (x, y) that lies on the solution curve. In this case, we are given that y = 7 when x = 0. This information is crucial because it allows us to determine the unique value of the constant of integration, C, that corresponds to the particular solution we are seeking. Finding the constant of integration is a key step in solving initial value problems, as it narrows down the infinite family of solutions to a single, specific function. To find C, we substitute the given values of x and y into the general solution we obtained in the previous step:
y = x³ - 3x² + 2x + C
Substituting x = 0 and y = 7, we get:
7 = (0)³ - 3(0)² + 2(0) + C
7 = 0 - 0 + 0 + C
7 = C
Therefore, the constant of integration, C, is equal to 7. This value is the key to unlocking the specific solution that satisfies both the differential equation and the initial condition. With the value of C in hand, we can now complete the final step of our problem.
Step 3: Substituting the Value of C to Obtain the Particular Solution
Now that we have determined the value of the constant of integration, C = 7, we can substitute this value back into the general solution we obtained in Step 1. This substitution will give us the particular solution, which is the unique function y(x) that satisfies both the given differential equation and the initial condition. The general solution was:
y = x³ - 3x² + 2x + C
Substituting C = 7, we get:
y = x³ - 3x² + 2x + 7
This is the particular solution to the given problem. It represents a specific cubic function that passes through the point (0, 7) and whose derivative is 3x² - 6x + 2. Obtaining the particular solution is the ultimate goal of solving initial value problems, as it provides a precise mathematical description of the function's behavior. We have successfully found y in terms of x, fulfilling the requirements of the problem.
Verification: Ensuring the Solution is Correct
Before we declare victory, it's always prudent to verify our solution. This involves two key checks: first, we need to ensure that our solution, y = x³ - 3x² + 2x + 7, satisfies the given differential equation, dy/dx = 3x² - 6x + 2. Second, we need to confirm that it satisfies the initial condition, y = 7 when x = 0. Verifying the solution is a crucial step in the problem-solving process, as it helps to catch any potential errors and ensure the accuracy of our results. To verify the differential equation, we differentiate our solution with respect to x:
dy/dx = d/dx (x³ - 3x² + 2x + 7)
Applying the power rule of differentiation, we get:
dy/dx = 3x² - 6x + 2
This matches the given differential equation, so our solution satisfies the first condition. To verify the initial condition, we substitute x = 0 into our solution:
y = (0)³ - 3(0)² + 2(0) + 7
y = 0 - 0 + 0 + 7
y = 7
This matches the given initial condition, so our solution satisfies the second condition as well. Since our solution satisfies both the differential equation and the initial condition, we can confidently conclude that it is the correct solution.
Conclusion: The Significance of Initial Value Problems
In conclusion, we have successfully found y in terms of x by integrating the given differential equation and using the initial condition to determine the constant of integration. The solution, y = x³ - 3x² + 2x + 7, represents a specific cubic function that satisfies both the equation and the condition. Understanding initial value problems is essential in many areas of science and engineering. These problems allow us to model and predict the behavior of systems over time, given some initial state. The ability to solve differential equations with initial conditions is a powerful tool that enables us to understand and manipulate the world around us. The process we have outlined in this exploration provides a clear and systematic approach to solving these types of problems, highlighting the fundamental role of integration in calculus and its applications.
