Homogeneous Differential Equations

Homogeneous Differential Equations: Your Key to JEE Success

Differential equations are a cornerstone of calculus and play a vital role in modeling real-world phenomena, from population growth to radioactive decay. For the JEE Main exam, mastering different types of differential equations is crucial, and Homogeneous Differential Equations are a frequent topic. Understanding how to identify, solve, and manipulate these equations will significantly boost your score.

What Makes an Equation "Homogeneous"?

Intuitively, a homogeneous differential equation is one where the ratio of $y$ to $x$ is the key. More formally, a first-order differential equation is homogeneous if it can be written in the form:

$$\frac{dy}{dx} = f\left(\frac{y}{x}\right)$$

This means the derivative $\frac{dy}{dx}$ is expressed as a function of the single variable $\frac{y}{x}$.

How to Spot a Homogeneous Function:

Consider a function $f(x, y)$. If, upon scaling both $x$ and $y$ by a factor of $\lambda$, the function transforms as:

then $f(x, y)$ is a homogeneous function of degree $n$. This is a more general definition. If you can rewrite your differential equation so that the right-hand side is a homogeneous function of degree 0, you've got a homogeneous differential equation!

Example: Consider $f(x, y) = x^2 + xy$. Then $f(\lambda x, \lambda y) = (\lambda x)^2 + (\lambda x)(\lambda y) = \lambda^2(x^2 + xy) = \lambda^2 f(x, y)$. This is homogeneous of degree 2.

The Magic Substitution: Turning Homogeneous into Separable

The trick to solving homogeneous equations lies in a clever substitution that transforms them into a separable form. Let's introduce a new variable $v$ such that:

$$v = \frac{y}{x}$$

This implies:

$$y = vx$$

Now, differentiate both sides of $y = vx$ with respect to $x$ using the product rule:

Why does this work? We're essentially changing our perspective. Instead of tracking $y$ and $x$ directly, we're tracking the ratio $v = y/x$ and how it changes with $x$.

Converting to Separable Form

Substitute $\frac{dy}{dx} = v + x\frac{dv}{dx}$ into the original homogeneous equation $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$. Since $v = \frac{y}{x}$, we have:

$$v + x\frac{dv}{dx} = f(v)$$

Rearrange to isolate the derivative term:

$$x\frac{dv}{dx} = f(v) - v$$

Now, we can separate the variables $v$ and $x$:

Aha! This is a separable differential equation! We now have all the $v$ terms on one side and all the $x$ terms on the other.

Solving and Back-Substitution

Integrate both sides of the separable equation:

$$\int \frac{dv}{f(v) - v} = \int \frac{dx}{x}$$

Solve the integrals. The right-hand side will be $\ln|x| + C$. The left-hand side might require partial fractions or other integration techniques. Once you have a solution in terms of $v$ and $x$, the final step is crucial: back-substitute $v = \frac{y}{x}$ to express the solution in terms of $y$ and $x$.

Example: Suppose after integrating, you get $\ln|v| = \ln|x| + C$. Then $v = kx$, where $k = e^C$. Back-substituting gives $\frac{y}{x} = kx$, or $y = kx^2$.

Tips for Cracking Homogeneous Equation Problems

• Always check for homogeneity first. Is the equation in the form $\frac{dy}{dx} = f(\frac{y}{x})$? Or, can you rewrite it to be in that form?
• Practice your integration skills. Solving the separable equation often requires a good command of integration techniques.
• Don't forget the constant of integration! Always include $+ C$ after integrating.
• Back-substitute meticulously. The final answer must be in terms of $x$ and $y$, not $v$.

Common Mistakes to Avoid

JEE-Specific Tricks

By understanding the theory, mastering the substitution technique, and practicing diligently, you'll be well-equipped to tackle homogeneous differential equation problems in the JEE Main exam. Good luck!