(x² + 4y² + 4xy)dy = (2x + 4y + 1)dx

How to solve this Differential equation??

To solve the differential equation \((x^2 + 4y^2 + 4xy)dy = (2x + 4y + 1)dx\), we can start by rearranging it into a more manageable form. This equation is a first-order differential equation, and we can express it in the standard form \(\frac{dy}{dx} = \frac{(2x + 4y + 1)}{(x^2 + 4y^2 + 4xy)}\).

Rearranging the Equation

First, let's rewrite the equation:

\( (x^2 + 4y^2 + 4xy) \frac{dy}{dx} = 2x + 4y + 1 \)

This gives us:

\( \frac{dy}{dx} = \frac{2x + 4y + 1}{x^2 + 4y^2 + 4xy} \)

Identifying the Type of Equation

This equation is not linear, but we can check if it can be simplified or if it has an integrating factor. A common approach is to see if it can be made separable or if it can be solved using an exact method.

Checking for Exactness

To determine if the equation is exact, we need to check if:

• Let \( M(x, y) = 2x + 4y + 1 \)
• Let \( N(x, y) = x^2 + 4y^2 + 4xy \)

We compute the partial derivatives:

• \( \frac{\partial M}{\partial y} = 4 \)
• \( \frac{\partial N}{\partial x} = 2x + 4y \)

Since \( \frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x} \), the equation is not exact.

Finding an Integrating Factor

Next, we can look for an integrating factor that depends on either \( x \) or \( y \). A common choice is to try \( \mu(y) \) or \( \mu(x) \). However, in this case, it might be more straightforward to manipulate the equation directly.

Using Substitution

Let's consider a substitution that might simplify the equation. One useful substitution in equations involving \( x \) and \( y \) is to express \( y \) in terms of \( x \) or vice versa. We can try \( v = \frac{y}{x} \), which gives us \( y = vx \) and \( dy = vdx + xdv \).

Substituting this into the equation gives:

\( (x^2 + 4(vx)^2 + 4x(vx)) (vdx + xdv) = (2x + 4(vx) + 1)dx \)

After simplification, we can separate variables or find a new form that is easier to integrate.

Integrating the Resulting Expression

After substituting and simplifying, we will arrive at a form that can be integrated. The integration will yield a solution in terms of \( x \) and \( y \). Depending on the complexity, we might need to use numerical methods or software tools for integration.

Final Steps

Once we have the integral, we can express the solution in the form \( F(x, y) = C \), where \( C \) is a constant. This gives us the implicit solution to the differential equation.

In summary, solving this differential equation involves rearranging it into a standard form, checking for exactness, possibly using substitutions, and integrating the resulting expression. Each step requires careful manipulation and understanding of differential equations, but with practice, it becomes more intuitive.