Solving The Differential Equation Y' = 5sin(x) - 5x^4 + 12e^(-4x) With Initial Condition

Hey guys! Today, we're diving into the fascinating world of differential equations! Specifically, we're going to tackle the problem of finding the solution to the differential equation y' = 5sin(x) - 5x^4 + 12e^(-4x), with the added condition that the solution passes through the point (0, 12). This means we're not just looking for any solution, but the particular solution that satisfies this initial condition. Think of it like finding the exact path a spaceship needs to take to reach a specific destination, not just any possible trajectory. So, buckle up and let's get started!

Understanding Differential Equations

Before we jump into the solution, let's take a moment to understand what a differential equation actually is. Simply put, a differential equation is an equation that relates a function to its derivatives. In our case, y' represents the first derivative of the function y with respect to x. The equation tells us how the rate of change of y depends on x. These equations are used everywhere in science and engineering to model everything from the motion of planets to the flow of heat to the growth of populations. They're like the secret language of the universe, describing how things change over time or space. The beauty of solving a differential equation is that we find the function y itself, which gives us a complete picture of the system being modeled. It's like going from knowing the speed of a car at every moment to knowing its exact position at any time.

The equation y' = 5sin(x) - 5x^4 + 12e^(-4x) is a first-order differential equation because it involves only the first derivative. It's also a separable equation, which means we can integrate both sides to find the solution. This is a common type of differential equation, and mastering the techniques to solve them is a fundamental skill in calculus and its applications. But don't worry, we'll break it down step by step so it's super easy to follow. We will walk through the integration, remembering our trig and power rules, and the special handling needed for that exponential term. The initial condition, passing through (0,12), will be our key to finding the specific solution, not just a general family of them. We'll plug in those values at the end to nail down the constant of integration. Think of it like tuning a radio to the right frequency – the initial condition helps us zero in on the one solution that fits our particular situation.

Step-by-Step Solution

Okay, let's get our hands dirty and solve this thing! The first step is to integrate both sides of the equation with respect to x. Remember, integration is the reverse process of differentiation, so it's like undoing the derivative to find the original function. Integrating y' gives us y, which is exactly what we're looking for. On the right side, we have a sum of three terms, so we can integrate each term separately. This makes the problem much more manageable. So, we have:

∫ y' dx = ∫ (5sin(x) - 5x^4 + 12e^(-4x)) dx

Now, let's tackle each term on the right side one by one. The integral of 5sin(x) is -5cos(x), remembering that the integral of sin(x) is -cos(x) and we just carry the constant 5 along for the ride. The integral of -5x^4 is -x^5, using the power rule for integration (add one to the exponent and divide by the new exponent). And finally, the integral of 12e^(-4x) is -3e^(-4x), remembering that the integral of e^(kx) is (1/k)e^(kx). Don't forget that crucial negative sign that comes from the -4 in the exponent! So, putting it all together, we get:

y = -5cos(x) - x^5 - 3e^(-4x) + C

Notice that we've added a constant of integration, C. This is super important! Remember, the derivative of a constant is always zero, so when we integrate, we lose that constant term. This means there are actually infinitely many solutions to the differential equation, each differing by a constant. The constant C represents this family of solutions. We need our initial condition to pinpoint the specific solution we're after. Think of C as a vertical shift – each value of C shifts the entire solution curve up or down. Our initial condition will tell us which of these curves passes through the point (0, 12).

Applying the Initial Condition

This is where the magic happens! We know that the solution passes through the point (0, 12). This means that when x = 0, y = 12. We can plug these values into our general solution to solve for the constant C. This is like finding the missing piece of the puzzle. Substituting x = 0 and y = 12 into our equation, we get:

12 = -5cos(0) - 0^5 - 3e^(-4*0) + C

Now, let's simplify. We know that cos(0) = 1 and e^0 = 1, so the equation becomes:

12 = -5(1) - 0 - 3(1) + C

12 = -5 - 3 + C

12 = -8 + C

Adding 8 to both sides, we find that C = 20. Hooray! We've found our constant of integration. This means we've pinpointed the specific solution that satisfies our initial condition. It's like finding the exact key that unlocks the solution to our problem.

The Particular Solution

Now that we've found the value of C, we can write down the particular solution to our differential equation. We simply substitute C = 20 back into our general solution:

y = -5cos(x) - x^5 - 3e^(-4x) + 20

And there you have it! This is the solution that satisfies both the differential equation and the initial condition. It's a beautiful blend of trigonometric, polynomial, and exponential functions, all working together to describe the behavior of y as a function of x. This is the path our spaceship needs to take, the precise trajectory that satisfies all the conditions. We can be confident in our answer because we followed a systematic approach, carefully integrating each term and using the initial condition to eliminate the ambiguity of the constant of integration. This solution represents a unique curve in the xy-plane, passing through the point (0, 12) and whose slope at any point is given by the original differential equation.

Verifying the Solution

It's always a good idea to double-check our work, just to be sure we haven't made any silly mistakes. We can verify our solution by taking its derivative and seeing if it matches the original differential equation. Let's differentiate our solution with respect to x:

y' = d/dx (-5cos(x) - x^5 - 3e^(-4x) + 20)

The derivative of -5cos(x) is 5sin(x). The derivative of -x^5 is -5x^4. The derivative of -3e^(-4x) is 12e^(-4x). And the derivative of the constant 20 is 0. So, we have:

y' = 5sin(x) - 5x^4 + 12e^(-4x)

This is exactly the same as the original differential equation! So, we know we've found the correct solution. We can also verify that our solution passes through the point (0, 12) by plugging in x = 0 and seeing if we get y = 12:

y(0) = -5cos(0) - 0^5 - 3e^(-4*0) + 20

y(0) = -5(1) - 0 - 3(1) + 20

y(0) = -5 - 3 + 20

y(0) = 12

This confirms that our solution does indeed satisfy the initial condition. We've not only found the solution, but we've also proven that it's correct! This feeling of validation is one of the great rewards of solving mathematical problems.

Conclusion

So, there you have it! We've successfully solved the differential equation y' = 5sin(x) - 5x^4 + 12e^(-4x), given that the solution passes through the point (0, 12). The particular solution is:

y = -5cos(x) - x^5 - 3e^(-4x) + 20

We walked through the steps carefully, from integrating the differential equation to applying the initial condition to verifying our solution. We also took the time to understand the underlying concepts, like what a differential equation is and why initial conditions are important. This wasn't just about getting the right answer; it was about understanding the process and building our mathematical skills. Solving differential equations can seem daunting at first, but by breaking them down into smaller steps and understanding the underlying principles, you can conquer even the most challenging problems. And remember, practice makes perfect! The more you solve these kinds of problems, the more comfortable and confident you'll become. Keep exploring, keep learning, and keep having fun with math!

Among the given options, the correct answer is:

c.) y = -5cos(x) - x^5 - 3e^(-4x) + 20