Variable Separable Method

Hello JEE aspirants! Differential equations might seem daunting, but they're a cornerstone of calculus and appear frequently in JEE Main. The Variable Separable Method is your first weapon in solving these equations. Master it, and you'll be well on your way to acing those problems!

What are Separable Equations?

Imagine you have a differential equation where you can neatly separate the terms involving $y$ and $dy$ on one side, and terms involving $x$ and $dx$ on the other. That's a separable equation! The beauty of this method lies in its simplicity: separate, integrate, and conquer.

Form: A differential equation is said to be variable separable if it can be written in the form:

$\frac{dy}{dx} = f(x) \cdot g(y)$

Here, $f(x)$ is a function of $x$ only, and $g(y)$ is a function of $y$ only.

Separating Variables: The Art of Rearrangement

The core idea is to isolate $y$ terms with $dy$ and $x$ terms with $dx$ . Think of it as organizing your study table – keeping similar items together. Starting with:

$\frac{dy}{dx} = f(x) \cdot g(y)$

We divide both sides by $g(y)$ (assuming $g(y) \neq 0$ ) and multiply both sides by $dx$ :

$\frac{dy}{g(y)} = f(x) \, dx$

Notice how we've successfully separated the variables. All $y$ terms are on the left, and all $x$ terms are on the right.

Integrating Both Sides: The Grand Finale

Once you've separated the variables, the next step is to integrate both sides of the equation. This gives you the general solution to the differential equation.

The Formula:

$\frac{dy}{dx} = f(x) \cdot g(y) \implies \int \frac{dy}{g(y)} = \int f(x) \, dx$

Let's break this down:

$\int \frac{dy}{g(y)}$ : This means integrating $\frac{1}{g(y)}$ with respect to $y$ . The result will be some function of $y$ , say $G(y)$ .
$\int f(x) \, dx$ : This means integrating $f(x)$ with respect to $x$ . The result will be some function of $x$ , say $F(x)$ .
Therefore, the general solution is: $G(y) = F(x) + C$ , where $C$ is the constant of integration.

Example: Consider the differential equation $\frac{dy}{dx} = x \cdot y$

Separate variables: $\frac{dy}{y} = x \, dx$
Integrate both sides: $\int \frac{dy}{y} = \int x \, dx$
Evaluate integrals: $\ln|y| = \frac{x^2}{2} + C$
Solve for y (optional, but often helpful): $y = e^{\frac{x^2}{2} + C} = e^{\frac{x^2}{2}} \cdot e^C = A e^{\frac{x^2}{2}}$ , where $A = e^C$ is another constant.

This gives us the general solution to the differential equation.

Tips for Solving Problems

Always check for $g(y) = 0$ solutions: Before dividing by $g(y)$ , make sure to check if $g(y) = 0$ gives a valid solution. In the example above, $y=0$ is a solution we would have missed had we not checked before dividing by $y$ .
Don't forget the constant of integration: Always add $+ C$ after integrating. This constant is crucial for the general solution.
Simplify after integrating: Try to simplify the expression as much as possible to make it easier to work with.
Solve for $y$ if possible: While not always necessary, solving for $y$ explicitly often makes the solution more useful.

Common Mistakes to Avoid

Forgetting the constant of integration: This is a very common mistake. Always remember to add $+ C$ after integrating.
Incorrectly separating variables: Make sure you separate the variables correctly before integrating. Double-check your algebra.
Not checking for $g(y) = 0$ solutions: As mentioned earlier, this can lead to missing solutions.
Mixing up integration rules: Be careful when applying integration rules. A small error can lead to a completely wrong answer.

JEE-Specific Tricks

While the variable separable method itself is straightforward, JEE problems often involve clever substitutions or require you to recognize the separable form within a more complex equation.

Look for hidden separable forms: Sometimes, a differential equation might not look separable at first glance. Try algebraic manipulations or substitutions to bring it into the separable form.
Use initial conditions to find particular solutions: JEE problems often provide initial conditions (e.g., $y(0) = 1$ ). Use these to find the specific value of the constant of integration $C$ and obtain the particular solution.
Practice, practice, practice: The more problems you solve, the better you'll become at recognizing separable equations and applying the method effectively.

Mastering the Variable Separable Method is crucial for tackling differential equations in JEE Main. Keep practicing, and you'll be solving these equations with confidence!