Solving a First-Order Differential Equation Using the Method of Separation of Variables

In this article, we will delve into the intricacies of solving a specific first-order differential equation of the form (2xy - 3 dx x^2 4y dy 0). We will explore how to transform the given differential equation into a form that can be solved using the method of separation of variables. Additionally, we will apply an initial condition to find the particular solution.

Problem Statement

The differential equation given is:

2xy - 3 dx x^2 4y dy 0

With the initial condition y1 2, we aim to find the solution to this differential equation.

Step-by-Step Solution

Step 1: Standard Form and Initial Setup

First, we rewrite the given differential equation in a more standard form:

2xy - 3 dx x^2 4y dy 0

This can be rearranged to:

dy/dx - (2xy - 3) / (x^2 4y)

This is a first-order ordinary differential equation. To solve it, we can use the method of separation of variables.

Step 2: Exactness Check

We check if the given equation is exact. We identify M(x, y) 2xy - 3 and N(x, y) x^2 4y. We compute the partial derivatives:

?M/?y 2x

?N/?x 2x

Since ?M/?y ?N/?x, the given differential equation is exact.

Step 3: Finding the Potential Function

Next, we find a function Ψ(x, y) such that:

?Ψ/?x M(x, y)

?Ψ/?y N(x, y)

Integrating M with respect to x:

Ψ(x, y) ∫ (2xy - 3) dx x^2y - 3x h(y)

Here, h(y) is an arbitrary function of y.

Now we differentiate Ψ(x, y) with respect to y:

?Ψ/?y x^2 h'(y)

Setting this equal to N:

x^2 h'(y) x^2 4y

This implies:

h'(y) 4y

Integrating h'(y) with respect to y:

h(y) 2y^2 C

Thus, the potential function is:

Ψ(x, y) x^2y - 3x 2y^2 C

Step 4: Setting the Potential Function to a Constant

The solution to the differential equation is given by setting Ψ(x, y) K, where K is a constant:

x^2y - 3x 2y^2 K

Step 5: Applying the Initial Condition

Now we apply the initial condition y1 2 when x 1:

1^2*2 - 3*1 2*2^2 K

2 - 3 8 K

K 7

Therefore, the particular solution is:

x^2y - 3x 2y^2 7

This represents the implicit solution to the differential equation.

Conclusion

In summary, we have solved the given first-order differential equation using the method of separation of variables. The implicit solution to the differential equation 2xy - 3 dx x^2 4y dy 0 with the initial condition y1 2 is: