(x^2+y^2-5)dx=(y+xy)dy

5 min read Jun 17, 2024

Solving the Differential Equation: (x^2 + y^2 - 5)dx = (y + xy)dy

This article will guide you through the process of solving the given differential equation:

(x^2 + y^2 - 5)dx = (y + xy)dy

This equation is classified as a non-linear, first-order differential equation. To solve it, we'll use the method of exact differential equations.

1. Checking for Exactness

A differential equation of the form M(x, y)dx + N(x, y)dy = 0 is exact if:

∂M/∂y = ∂N/∂x

Let's rearrange our given equation:

(x^2 + y^2 - 5)dx - (y + xy)dy = 0

Now, we identify M(x, y) and N(x, y):

M(x, y) = x^2 + y^2 - 5
N(x, y) = - (y + xy)

Let's calculate the partial derivatives:

∂M/∂y = 2y
∂N/∂x = -y

Since ∂M/∂y ≠ ∂N/∂x, the given equation is not exact.

2. Finding an Integrating Factor

To make the equation exact, we need to find an integrating factor, µ(x, y).

For an equation of the form M(x, y)dx + N(x, y)dy = 0, if:

(∂M/∂y - ∂N/∂x) / N is a function of x only, then µ(x) = exp(∫[(∂M/∂y - ∂N/∂x)/N]dx)
(∂N/∂x - ∂M/∂y) / M is a function of y only, then µ(y) = exp(∫[(∂N/∂x - ∂M/∂y)/M]dy)

Let's calculate (∂M/∂y - ∂N/∂x) / N:

[(2y - (-y)) / (-y - xy)] = -3/(1 + x)

This is a function of x only. Therefore, our integrating factor is:

µ(x) = exp(∫[-3/(1 + x)]dx) = exp(-3ln(1 + x)) = (1 + x)^-3

3. Multiplying the Equation by the Integrating Factor

Multiply both sides of the original equation by µ(x):

(1 + x)^-3(x^2 + y^2 - 5)dx - (1 + x)^-3(y + xy)dy = 0

Now, let's check if this equation is exact:

M(x, y) = (1 + x)^-3(x^2 + y^2 - 5)
N(x, y) = -(1 + x)^-3(y + xy)
∂M/∂y = 2y(1 + x)^-3
∂N/∂x = 2y(1 + x)^-3

We see that ∂M/∂y = ∂N/∂x. Therefore, the equation is now exact.

4. Solving the Exact Equation

For an exact equation, there exists a function F(x, y) such that:

∂F/∂x = M(x, y)
∂F/∂y = N(x, y)

Let's integrate ∂F/∂x = M(x, y) with respect to x:

F(x, y) = ∫[(1 + x)^-3(x^2 + y^2 - 5)]dx = -1/2(1 + x)^-2(x^2 + y^2 - 5) + g(y)

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

Now, differentiate F(x, y) with respect to y and equate it to N(x, y):

∂F/∂y = -(1 + x)^-2y + g'(y) = -(1 + x)^-3(y + xy)

Solving for g'(y), we get:

g'(y) = 0

Integrating g'(y), we obtain:

g(y) = C

Where C is an arbitrary constant.

Finally, the general solution to the differential equation is:

F(x, y) = -1/2(1 + x)^-2(x^2 + y^2 - 5) + C = 0

This implicit equation defines the solution for the given differential equation.