Animated Solution for Mathematics - Differential Equations: The order and degree of the differential equation $(1+3\frac{dy}{dx}{)}^{2/3}=4\frac{d^{3}y}{d{x}^3}$ are

Understanding Order and Degree

Welcome, future engineers! Today, we are going to demystify a concept that often acts as a gatekeeper in JEE Advanced exams: the order and degree of a differential equation.

It is a topic that seems simple on the surface, but it is riddled with subtle traps designed to test your conceptual clarity. Let us dive into the equation:

The Order - The Highest Peak

Think of the order of a differential equation as the highest mountain peak in a landscape. It is the most straightforward part of the analysis.

The order is simply the order of the highest derivative present in the equation. It does not care about coefficients, it does not care about powers, and it certainly does not care about radicals.

When we scan our equation, we see two derivatives: $\frac{dy}{dx}$ (the first derivative) and $\frac{d^{3}y}{d{x}^3}$ (the third derivative). Clearly, the third derivative is the highest. Therefore, the order is 3.

The Degree - The Hidden Trap

Now, we arrive at the degree. This is where the JEE examiners love to hide their traps.

The definition of degree is the power of the highest-order derivative, but with a critical condition: the equation must be a polynomial in its derivatives. This means no fractional powers, no square roots, and no cube roots on any derivative term.

If you look at our equation, we have a fractional power of $2/3$ on the left side. If you were to rush and say the degree is 1 because the third derivative on the right side appears to have a power of 1, you would fall right into the trap. We must first perform 'algebraic surgery' to rationalize the equation.

The Rationalization - Algebraic Surgery

To make the equation a polynomial in derivatives, we need to eliminate that denominator of 3 in the exponent $2/3$. The most effective tool in our arsenal is to raise both sides of the equation to the power of 3.

Let us perform this operation:

On the left side, the power of 3 cancels the denominator, leaving us with $(1+3\frac{dy}{dx}{)}^2$. On the right side, we have $4^3\cdot (\frac{d^{3}y}{d{x}^3}{)}^3$, which simplifies to $64(\frac{d^{3}y}{d{x}^3}{)}^3$.

Now, look at the equation:

It is clean, it is rational, and it is a polynomial in derivatives.

The Conclusion

With the equation rationalized, we can finally identify the degree. We look at our highest-order derivative, $\frac{d^{3}y}{d{x}^3}$, and observe its power in this new, rationalized form.

The power is 3. Thus, the degree is 3.

We have successfully navigated the trap! The order is 3, and the degree is 3. The correct answer is the coordinate pair $(3,3)$.

Remember, in the world of JEE, always check for fractional powers on derivatives before declaring the degree. Stay curious, keep practicing, and I will see you in the next challenge!