Dy Dx Xy 3x Y 3 Xy 2x 4y 8

Understanding and Solving the Differential Equation dy/dx = xy + 3xy + 3xy + 2x + 4y + 8

The equation dy/dx = xy + 3xy + 3xy + 2x + 4y + 8 presents a complex yet solvable problem in the realm of differential equations. Which means at first glance, the expression may seem overwhelming due to its repetitive terms and mixed variables. That said, by simplifying and analyzing the structure, we can uncover a clear path to solving it. This article will guide you through the process of interpreting, simplifying, and solving this equation, while also explaining the underlying mathematical principles. Whether you’re a student grappling with calculus or a learner seeking to deepen your understanding of differential equations, this explanation will provide the tools and insights needed to tackle similar problems.

Introduction: Breaking Down the Equation

The equation dy/dx = xy + 3xy + 3xy + 2x + 4y + 8 is a first-order ordinary differential equation (ODE). Because of that, the left-hand side, dy/dx, represents the derivative of y with respect to x, while the right-hand side is a combination of terms involving x and y. The presence of multiple xy terms suggests that the equation can be simplified by combining like terms. This simplification is crucial for identifying the type of differential equation we are dealing with and applying the appropriate solution method. Nothing fancy.

By combining the xy terms, we get:
dy/dx = (1 + 3 + 3)xy + 2x + 4y + 8
dy/dx = 7xy + 2x + 4y + 8

This simplified form reveals that the equation is a linear differential equation. Linear ODEs have a standard solution approach, which we will explore in detail. The key to solving such equations lies in recognizing their structure and applying the integrating factor method.

Steps to Solve the Differential Equation

Solving a linear differential equation involves several systematic steps. Let’s walk through the process for dy/dx = 7xy + 2x + 4y + 8.

Step 1: Rewrite the Equation in Standard Form

The standard form of a linear first-order ODE is:
dy/dx + P(x)y = Q(x)

To convert our equation into this form, we need to isolate the y terms on one side. Starting with:
*dy/dx = 7xy +

Step 1: Rewrite the Equation in Standard Form (Continued)

dy/dx = 7xy + 4y + 2x + 8
Now, factor out the y term from the first two terms:
dy/dx + 4y = 7xy + 2x + 8
Next, rearrange the equation to match the standard form:
dy/dx + 4y = 7xy + 2x + 8

Step 2: Identify P(x) and Q(x)

Comparing our equation to the standard form dy/dx + P(x)y = Q(x), we can identify:
P(x) = 4
Q(x) = 7xy + 2x + 8

Step 3: Calculate the Integrating Factor

The integrating factor, denoted by μ(x), is calculated as:
μ(x) = e^(∫P(x) dx)
In our case, P(x) = 4, so:
μ(x) = e^(∫4 dx) = e^(4x)

Step 4: Multiply the Entire Equation by the Integrating Factor

Multiply both sides of the standard form equation by e^(4x):
e^(4x)(dy/dx + 4y) = e^(4x)(7xy + 2x + 8)
This simplifies to:
e^(4x)dy/dx + 4e^(4x)y = 7xye^(4x) + 2xe^(4x) + 8e^(4x)

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Step 5: Recognize the Left-Hand Side as a Derivative

The left-hand side of the equation is now the derivative of the product y and the integrating factor e^(4x):
(ye^(4x))' = 7xye^(4x) + 2xe^(4x) + 8e^(4x)

Step 6: Integrate Both Sides with Respect to x

Integrate both sides with respect to x:
∫(ye^(4x))' dx = ∫(7xye^(4x) + 2xe^(4x) + 8e^(4x)) dx
ye^(4x) = ∫(7xye^(4x) + 2xe^(4x) + 8e^(4x)) dx
To solve the integral on the right-hand side, we can use integration by parts. Let's integrate the first term, ∫7xye^(4x) dx. We'll use u = 7xy and dv = e^(4x) dx. Then du = 7y dx and v = (1/4)e^(4x).
∫7xye^(4x) dx = (7xy/4)e^(4x) - ∫(7y/4)e^(4x) dx = (7xy/4)e^(4x) - (7y/4)(1/4)e^(4x) + C1 = (7xy/4)e^(4x) - (7y/16)e^(4x) + C1.
The integral of the second term is ∫2xe^(4x) dx. Using integration by parts with u = 2x and dv = e^(4x) dx, we get du = 2 dx and v = (1/4)e^(4x).
∫2xe^(4x) dx = (2x/4)e^(4x) - ∫(2/4)e^(4x) dx = (x/2)e^(4x) - (1/2)(1/4)e^(4x) + C2 = (x/2)e^(4x) - (1/8)e^(4x) + C2.
The integral of the third term is ∫8e^(4x) dx = 2e^(4x) + C3.
Combining these results, we get:
ye^(4x) = [(7xy/4)e^(4x) - (7y/16)e^(4x) + C1] + [(x/2)e^(4x) - (1/8)e^(4x) + C2] + [2e^(4x) + C3]
ye^(4x) = (7/4)xye^(4x) + (x/2)e^(4x) - (7/16)ye^(4x) + 2e^(4x) + C (where C = C1 + C2 + C3).

Step 7: Solve for y

Divide both sides by e^(4x):
y = (7/4)x + x/2 - (7/16)y + 2e^(-4x) + Ce^(-4x)
Now, isolate the y terms:
y + (7/16)y = (7/4)x + x/2 + 2e^(-4x) + Ce^(-4x)

(23/16)y = (7/4)x + (1/2)x + 2e^(-4x) + Ce^(-4x)
(23/16)y = (7/4 + 2/4)x + 2e^(-4x) + Ce^(-4x)
(23/16)y = (9/4)x + 2e^(-4x) + Ce^(-4x)
Finally, solve for y:
y = (16/23) * [(9/4)x + 2e^(-4x) + Ce^(-4x)]
y = (16/23) * (9/4)x + (16/23) * 2e^(-4x) + (16/23) * Ce^(-4x)
y = (36/23)x + (32/23)e^(-4x) + (16C/23)e^(-4x)
Let K = 16C/23, which is another arbitrary constant.
y = (36/23)x + (32/23)e^(-4x) + Ke^(-4x)

Conclusion

We have successfully solved the first-order linear differential equation dy/dx = 7xy + 4y + 2x + 8. By systematically applying the method of integrating factors, we first rewrote the equation in standard form, identified P(x) and Q(x), calculated the integrating factor, multiplied the equation by the integrating factor, recognized the resulting derivative, integrated both sides, and finally solved for y. The general solution is given by:

y = (36/23)x + (32/23)e^(-4x) + Ke^(-4x)

where K is an arbitrary constant determined by any initial conditions provided. This demonstrates the power of the integrating factor method in solving a wide range of first-order linear differential equations, providing a structured approach to finding solutions that would be difficult or impossible to obtain through other methods. The key is recognizing the form of the equation and applying the steps in a logical and consistent manner.