Solving Differential Equations: dy/dx 2xy1 2 and dy/dx 4xy1 2

Introduction

In this article, we will explore how to solve two specific types of first-order differential equations: dy/dx 2xy1 2 and dy/dx 4xy1 2. These equations are common in various fields of mathematics and engineering, and understanding how to solve them can be crucial for many applications.

Solving dy/dx 2xy1 2

To solve the differential equation dy/dx 2xy1 2, we can use the method of integrating factors. Let's break down the process step by step.

Step 1: Setup and Integrating Factors

We start with the equation:

dy/dx - 4x - 2y - 1 0

or rearranged as:

dy/dx 4x 2y 1

Here, the coefficient of y is 2. The integrating factor u(x) is given by:

u(x) e∫2 dx e2x

Step 2: Multiply by the Integrating Factor

Multiplying both sides of the equation by e2x, we get:

e2xdy/dx 2e2xy 2e2xx e2x

The left-hand side is now the derivative of the product y·e2x with respect to x:

d/dx(ye2x) 2e2xx e2x

Step 3: Integrate Both Sides

Integrate both sides with respect to x:

ye2x ∫(2e2xx e2x) dx

The integral on the right-hand side can be solved using integration by parts or by recognizing it as a standard form:

ye2x e2x(x - 1/2) C

Step 4: Solve for y

Dividing both sides by e2x, we obtain:

y (x - 1/2) Ce-2x

Solving dy/dx 4xy1 2

For the second differential equation, dy/dx 4xy1 2, we can use a substitution method.

Step 1: Substitution

Let u 4x - y - 1. Then, du/dx 4 - dy/dx, and hence:

dy/dx 4 - du/dx

Substitute into the differential equation:

4 - (du/dx) 4u2

du/dx - 4u2 -4

Step 2: Separation of Variables

Solve the separated form:

d/dx(u-1) 4x

Integrate both sides:

u-1 2x C

Step 3: Solve for y

Solving for u and substituting back for u 4x - y - 1:

1/(2x C) 4x - y - 1

y 4x - 1 - 1/(2x C)

Conclusion

In this article, we have discussed methods for solving two specific types of first-order differential equations. We used integrating factors to solve dy/dx 2xy1 2 and a substitution method for dy/dx 4xy1 2. Understanding these techniques can greatly enhance your problem-solving skills in various fields of mathematics and related sciences.

Keywords: differential equations, solve dy/dx, integrating factors, substitution method