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

7 min read Jun 17, 2024

Solving the Differential Equation (y^2 + xy^3)dx + (5y^2 - xy + y^3sin(y))dy = 0

This article explores the solution to the given differential equation, which is a non-exact differential equation. We will use the method of finding an integrating factor to transform the equation into an exact one, making it solvable.

Identifying the Equation Type

The given differential equation is of the form:

M(x, y)dx + N(x, y)dy = 0

Where:

M(x, y) = y^2 + xy^3
N(x, y) = 5y^2 - xy + y^3sin(y)

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

∂M/∂y = ∂N/∂x

Let's calculate the partial derivatives:

∂M/∂y = 2y + 3xy^2
∂N/∂x = -y

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

Finding an Integrating Factor

To make the equation exact, we need to find an integrating factor, denoted by μ(x, y). This factor, when multiplied with the original equation, will make it exact. There are two common approaches:

μ(x) = e^(∫(∂N/∂x - ∂M/∂y)/M dy) if the expression ((∂N/∂x - ∂M/∂y)/M) is a function of y only.
μ(y) = e^(∫(∂M/∂y - ∂N/∂x)/N dx) if the expression ((∂M/∂y - ∂N/∂x)/N) is a function of x only.

Let's apply these approaches to our equation:

((∂N/∂x - ∂M/∂y)/M) = (-y - 2y - 3xy^2) / (y^2 + xy^3) = (-3y - 3xy^2) / (y^2 + xy^3). This is not a function of y only.
((∂M/∂y - ∂N/∂x)/N) = (2y + 3xy^2 + y) / (5y^2 - xy + y^3sin(y)) = (3y + 3xy^2) / (5y^2 - xy + y^3sin(y)). This is also not a function of x only.

Therefore, neither approach directly provides an integrating factor. However, we can observe that the expression ((∂N/∂x - ∂M/∂y)/M) can be simplified by factoring out -3y from the numerator. This suggests a potential integrating factor of the form μ(y) = 1/y^3.

Let's verify if this is indeed an integrating factor:

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

Now, we need to check if ∂(μ(y)M(x, y))/∂y = ∂(μ(y)N(x, y))/∂x:

∂(μ(y)M(x, y))/∂y = -1/y^2
∂(μ(y)N(x, y))/∂x = -1/y^2

As the partial derivatives are equal, μ(y) = 1/y^3 is indeed an integrating factor.

Solving the Exact Equation

Now, the equation:

(1/y + x)dx + (5/y - x/y^2 + sin(y))dy = 0

is exact. To solve it, we need to find a function ψ(x, y) such that:

∂ψ/∂x = 1/y + x
∂ψ/∂y = 5/y - x/y^2 + sin(y)

Integrating the first equation with respect to x, we get:

ψ(x, y) = x/y + x^2/2 + h(y)

where h(y) is an arbitrary function of y.

Now, differentiating this expression with respect to y and equating it to the second equation:

∂ψ/∂y = -x/y^2 + h'(y) = 5/y - x/y^2 + sin(y)

This implies:

h'(y) = 5/y + sin(y)

Integrating this equation with respect to y:

h(y) = 5ln(y) - cos(y) + C

where C is an arbitrary constant.

Therefore, the general solution to the original differential equation is:

ψ(x, y) = x/y + x^2/2 + 5ln(y) - cos(y) + C = 0

This solution implicitly defines the relationship between x and y that satisfies the original differential equation.

Conclusion

By finding the integrating factor and transforming the non-exact equation into an exact one, we were able to solve the differential equation (y^2 + xy^3)dx + (5y^2 - xy + y^3siny)dy = 0, arriving at the general solution x/y + x^2/2 + 5ln(y) - cos(y) + C = 0. This process demonstrates the powerful technique of integrating factors in solving non-exact differential equations.