The degree of the differential equation $(\frac{d^{3}y}{dx^{3}})+x^{2}(\frac{d^{2}y}{dx^{2}})^{3}+(\frac{dy}{dx})^{5} +y^{7}=0$ is

Question

Accepted Answer

The degree of a differential equation is defined as the highest power of the highest order derivative present in the equation, provided the equation is a polynomial in its derivatives. Let's analyze the given differential equation:

$$
\left(\frac{d^{3}y}{dx^{3}}\right) + x^{2}\left(\frac{d^{2}y}{dx^{2}}\right)^{3} + \left(\frac{dy}{dx}\right)^{5} + y^{7} = 0
$$

1. **Identify the highest order derivative**: The highest order derivative in this equation is $\frac{d^{3}y}{dx^{3}}$.

2. **Check if the equation is a polynomial in its derivatives**: The equation is indeed a polynomial in its derivatives because each term involving derivatives is raised to a power, and there are no transcendental functions (like sine, cosine, exponential, etc.) of the derivatives.

3. **Determine the degree**: The degree of the differential equation is determined by the highest power of the highest order derivative. In this equation, the highest order derivative is $\frac{d^{3}y}{dx^{3}}$, and it appears to the power of 1. Therefore, the degree of the differential equation is 1.

Now, let's explain why the other options are incorrect:

- **Option 2 (Degree 3)**: This option might be mistakenly considered if one looks at the term $(\frac{d^{2}y}{dx^{2}})^{3}$. However, the degree of the differential equation is determined by the highest order derivative, not the highest power of any derivative. Since $\frac{d^{3}y}{dx^{3}}$ is the highest order derivative and it is raised to the power of 1, the degree is not 3.

- **Option 3 (Degree 5)**: This option could be incorrectly chosen by considering the term $(\frac{dy}{dx})^{5}$. However, the degree is not determined by the highest power of any derivative, but by the highest power of the highest order derivative. Since $\frac{d^{3}y}{dx^{3}}$ is the highest order derivative and it is raised to the power of 1, the degree is not 5.

- **Option 4 (Degree 7)**: This option might be mistakenly considered by looking at the term $y^{7}$. However, the degree of a differential equation is concerned with the derivatives, not the powers of the function $y$ itself. Since the highest order derivative $\frac{d^{3}y}{dx^{3}}$ is raised to the power of 1, the degree is not 7.

In conclusion, the correct degree of the differential equation is 1, as supported by the analysis above. Therefore, Option 1 is indeed the correct answer.