Q.Find the particular solution of the differential equation $\log\left(\frac{dy}{dx}\right) = 3x + 4y$ given that $y = 0$ when $x = 0$.

Separation of Variables: From Intuition to Precision

Imagine you're baking a cake. The recipe says "mix the dry ingredients separately, then add the wet ones." You keep things that belong together together, and things that don't apart — until the right moment. Separation of Variables does exactly that for certain kinds of equations.

The Core Intuition

Some equations involve two different kinds of change happening at once. Think of a cup of hot coffee cooling down. The rate at which it cools depends on:

• The temperature difference between the coffee and the room (a function of time)
• The surface area of the cup (a function of shape, not time)

These two influences are tangled together in one equation. Separation of Variables is the mathematical trick that untangles them — it lets you handle the time part first, then the space part separately.

The Precise Statement

Separation of Variables applies to ordinary differential equations (ODEs) of the form:

where the right-hand side is a product of a function of $x$ alone and a function of $y$ alone. The method works in three clean steps:

Step 1: Separate. Multiply both sides by $dx$ and divide by $g(y)$ (assuming $g(y) \neq 0$). This moves all $y$'s to one side and all $x$'s to the other.

Step 2: Integrate. Put an integral sign on both sides:

Step 3: Solve. Evaluate both integrals and solve for $y$ explicitly if possible.

Why This Works

The justification is the chain rule in reverse. From $\frac{dy}{dx} = f(x)g(y)$, rewrite it as:

Now integrate both sides with respect to $x$:

The left side is a substitution waiting to happen: $\frac{dy}{dx} dx = dy$, so you get $\int \frac{1}{g(y)} dy$. That's the entire trick — the chain rule dressed up.

A Concrete Example

Solve $\frac{dy}{dx} = 2xy$, with $y(0) = 3$.

Step 1: Separate. Divide both sides by $y$ (assuming $y \neq 0$):

Step 2: Integrate.

Step 3: Solve for $y$.

Let $A = \pm e^C$ (absorbing the absolute value): $y = A e^{x^2}$. Now use $y(0) = 3$: $3 = A e^{0} = A$, so $A = 3$.

Final answer: $\boxed{y = 3e^{x^2}}$

When Does It Apply?

Separation of Variables works for first-order ODEs of the form $\frac{dy}{dx} = f(x)g(y)$, and for certain partial differential equations like the heat equation (a more advanced use — the idea is the same: assume the solution is a product of functions of each variable). It does not work for equations where $x$ and $y$ are added, subtracted, or composed in non-product ways, or for higher-order ODEs (generally).

The Big Picture

Separation of Variables is your first real tool for solving differential equations. It reduces a problem with two moving parts into two independent single-variable integrals: you separate the variables, handle each alone, then reassemble with the initial condition. Think of it as untangling a knot by pulling the two ends apart — once separate, each piece is easy to deal with.

Separation of Variables is the very first solving technique taught in NCERT's Class 12 Differential Equations chapter and one of the most heavily tested skills in CBSE board exams and JEE Main. Students searching "separable differential equations important questions" or "differential equations class 12 formula" will find this method the starting point for almost every other technique in the chapter.