Solving Differential Equations: The General Solution of y - 4y x

In this article, we will provide a detailed explanation of a differential equation commonly encountered in advanced mathematics and physics. The equation in question is y - 4y x. We will delve into the step-by-step process of finding its general solution, utilizing the concept of an integrating factor.

Understanding the Equation

The equation y - 4y x can be simplified to -3y x, but this does not fully capture the complexity of the problem. In the context of differential equations, this is a first-order linear differential equation. The goal is to find a function y(x) that satisfies this equation. To do this, we use the method of integrating factors.

Integrating Factor Method

The integrating factor method is a powerful technique to solve first-order linear differential equations. The general form of a first-order linear differential equation is:

y' P(x)y Q(x)

In our case, the equation can be rewritten as:

y' - 4y x

Here, P(x) -4 and Q(x) x. The integrating factor is given by μ(x) e^{∫P(x)dx}.

Step-by-Step Solution

1. **Compute the Integrating Factor:**

The integrating factor is:

μ(x) e^{∫-4dx} e^{-4x}

2. **Multiply Both Sides by the Integrating Factor:**

Multiplying both sides of the original equation by e^{-4x}, we get:

e^{-4x}y - 4e^{-4x}y xe^{-4x}

3. **Apply the Product Rule on the Left Side:**

The left side can be rewritten using the product rule:

d/dx(e^{-4x}y) xe^{-4x}

4. **Integrate Both Sides:**

Integrating both sides with respect to x, we get:

∫ d(e^{-4x}y) ∫ xe^{-4x} dx

5. **Solve the Integral on the Right Side:**

Using integration by parts, we can solve the integral on the right side:

∫ xe^{-4x} dx -1/4∫ x-4e^{-4x} dx

-1/4∫ x de^{-4x}

-1/4[xe^{-4x} - ∫ e^{-4x} dx]

-1/4[xe^{-4x} 1/4e^{-4x}]

-1/4xe^{-4x} - 1/16e^{-4x}

6. **Combine the Results:**

Thus, we have:

e^{-4x}y -1/16 e^{-4x} (4x 1) C

7. **Solve for y(x):

Multiplying both sides by e^{4x}, we get the general solution:

y(x) Ce^{4x} - 1/16 (4x 1)

General Solution Recap

The general solution to the differential equation y - 4y x is:

y(x) Ce^{4x} - 1/16 (4x 1)

where C is an arbitrary constant determined by initial or boundary conditions.

Conclusion

Hello, if you found this article interesting and are curious about the steps, it's a good idea to practice yourself. Solving differential equations is a fundamental skill in many fields of science and engineering. If you're just learning, you'll find that the more you practice, the better you'll understand the concepts and techniques involved.

Further Readings and References

For more insights into differential equations and related topics, you might want to explore:

- Elementary Differential Equations and Boundary Value Problems by William E. Boyce and Richard C. DiPrima. - Differential Equations: Techniques, Theory, and Applications by Robert L. Borrelli and Colin B. Pierce.

Keep learning and exploring, and you'll find that differential equations open the door to a wealth of mathematical and physical applications.