Solving Differential Equations Using Separation of Variables

Differential equations often arise in various fields such as physics, engineering, and mathematics. Solving these equations can give us valuable insights into the behavior of systems, but the methods can vary widely depending on the specific form of the equation. This article focuses on the method of separation of variables, a powerful technique that can simplify many differential equations into more manageable forms.

Introduction to Separation of Variables

The key idea behind the method of separation of variables is to rewrite a differential equation in a form that allows us to separate the variables on each side of the equation. For a first-order differential equation of the form fy dy gx dx, we can integrate both sides independently to find the solution. This process simplifies the problem and often leads to tractable results.

Step-by-Step Guide to Solving Using Separation of Variables

Let's go through the steps to solve a differential equation using the method of separation of variables. We will consider the equation:

x frac{dy}{dx} y^3 - y

The first step is to manipulate the equation to separate the variables. We divide both sides by xy^3 - y to obtain:

frac{dy}{y^3 - y} frac{dx}{x}

This form allows us to separate the variables, leading to two independent integrals:

int frac{dy}{y^3 - y} int frac{dx}{x}

Integrating both sides, we have:

ln y - frac{ln(y^2 1)}{2} ln x C

By noting that ln A - ln B lnfrac{A}{B} and nln A ln A^n, we can rewrite the equation as:

ln frac{y}{sqrt{y^2 1}} ln (x C)

Exponentiating both sides, we obtain:

frac{y}{sqrt{y^2 1}} kx

where k is some arbitrary constant. Solving for y^2, we get:

y^2 frac{x^2}{k^2 x^2 - 1}

Taking the square root, we have:

y pmfrac{x}{sqrt{k^2 x^2 - 1}}

Solution as a Bernoulli Equation

Another approach to solving the same differential equation is to consider it as a Bernoulli Equation. This form allows for a more straightforward transformation and solution process. The original equation:

xfrac{dy}{dx} y^3 - y

can be written in the form:

y^{-3}y' - frac{1}{x}y^{-2} frac{1}{x}

By substituting u y^{-2} and u' -2y^{-3}y', the equation transforms to:

-frac{1}{2}u' - frac{1}{x}u frac{1}{x}

This is a first-order linear differential equation, which can be solved using methods for such equations. The integrating factor is:

m e^{int frac{2}{x} dx} x^2

Multiplying through by the integrating factor, we get:

(x^2u)' -2x

Integrating both sides, we obtain:

x^2u -x^2C

and thus:

u frac{C - x^2}{x^2}

Since y^2 frac{1}{u}, we have:

y^2 frac{x^2}{C - x^2}

Therefore, the solution is:

y pm frac{x}{sqrt{C - x^2}}

Finally, note that y 0 is also a solution.