Understanding the Degree of a Differential Equation

The question asks for the degree of the given differential equation: \(1+\left(\frac{dy}{dx}\right)^2 =\left(\frac{d^2y}{dx^2}\right)^{\frac{4}{3}}\).

To find the degree of a differential equation, we first need to determine its order. The order of a differential equation is the order of the highest derivative present in the equation.

In the given equation, the derivatives present are \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\).

• \(\frac{dy}{dx}\) is the first derivative, so its order is 1.
• \(\frac{d^2y}{dx^2}\) is the second derivative, so its order is 2.

The highest order derivative is \(\frac{d^2y}{dx^2}\), which has an order of 2. Therefore, the order of this differential equation is 2.

Now, to find the degree, we need to look at the power of the highest order derivative after the equation has been made free from radicals and fractions as far as derivatives are concerned.

The given equation is: \[1+\left(\frac{dy}{dx}\right)^2 =\left(\frac{d^2y}{dx^2}\right)^{\frac{4}{3}}\] Notice the term \(\left(\frac{d^2y}{dx^2}\right)^{\frac{4}{3}}\). The exponent \(\frac{4}{3}\) represents a fractional power, which is equivalent to a cube root raised to the power of 4. To eliminate this fractional power involving the derivative \(\frac{d^2y}{dx^2}\), we need to cube both sides of the entire equation.

Cubing both sides, we get: \[\left(1+\left(\frac{dy}{dx}\right)^2\right)^3 = \left(\left(\frac{d^2y}{dx^2}\right)^{\frac{4}{3}}\right)^3\]

Simplifying the right side: \[\left(\left(\frac{d^2y}{dx^2}\right)^{\frac{4}{3}}\right)^3 = \left(\frac{d^2y}{dx^2}\right)^{\frac{4}{3} \times 3} = \left(\frac{d^2y}{dx^2}\right)^4\]

So, the equation becomes: \[\left(1+\left(\frac{dy}{dx}\right)^2\right)^3 = \left(\frac{d^2y}{dx^2}\right)^4\]

In this simplified equation, which is free from fractional powers involving derivatives, the highest order derivative is still \(\frac{d^2y}{dx^2}\).

The power of this highest order derivative (\(\frac{d^2y}{dx^2}\)) is 4.

Therefore, the degree of the differential equation is 4.

Step-by-Step Determination of Degree

1. Identify the highest order derivative in the equation. It is \(\frac{d^2y}{dx^2}\).
2. Identify if there are any fractional or radical powers involving derivatives. Yes, the term \(\left(\frac{d^2y}{dx^2}\right)^{\frac{4}{3}}\).
3. Raise both sides of the equation to a power that eliminates the fractional exponent for the derivative. In this case, the denominator of the exponent is 3, so cube both sides.
4. Simplify the equation after raising to the appropriate power: \(\left(1+\left(\frac{dy}{dx}\right)^2\right)^3 = \left(\frac{d^2y}{dx^2}\right)^4\).
5. Identify the highest order derivative again in the cleared equation. It is still \(\frac{d^2y}{dx^2}\).
6. The power of this highest order derivative is the degree of the differential equation. The power of \(\frac{d^2y}{dx^2}\) is 4.

Thus, the degree is 4.

Revision Table: Key Concepts for Differential Equations

Additional Information on Order and Degree

Understanding the order and degree of a differential equation is fundamental in classifying and solving them.

• Order: This is always a positive integer. It directly tells you the 'level' of differentiation involved. A first-order equation involves only \(\frac{dy}{dx}\), a second-order involves \(\frac{d^2y}{dx^2}\) and possibly lower derivatives, and so on.
• Degree: This is also a positive integer. However, the degree is defined only if the differential equation can be written as a polynomial in terms of the derivatives. If a differential equation contains terms like \(\sin(\frac{dy}{dx})\), \(e^{\frac{d^2y}{dx^2}}\), or \(\log(\frac{dy}{dx})\), its degree is undefined because it cannot be expressed as a polynomial in derivatives. In our case, after clearing the fractional exponent, the equation becomes a polynomial in terms of the derivatives \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\), so the degree is well-defined.

Always remember to clear any fractional or radical powers that involve the derivatives before determining the degree. Only consider the power of the highest order derivative after this step.