Question

Find the order and degree of the following differential equations: (i) \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}=\left\\{1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{4}\right\\}^{\frac{5}{3}}\) (ii) \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+3\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}=\mathrm{x} \ell \mathrm{n} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\) (iii) \(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{x} \ell \mathrm{n}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)\) (iv) \(\left(\frac{d^{4} y}{d x^{4}}\right)^{3}+4\left(\frac{d y}{d x}\right)^{7}+6 y=5 \cos 3 x\).

Short answer

Based on the provided step-by-step solution, the question for students could be: Question: Determine the order and degree for the given differential equations: (i) $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d}\mathrm{x}^{2}}=\left\\{1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{4}\right\\}^{\frac{5}{3}}$ (ii) $\frac{\mathrm{d}^{2}\mathrm{y}}{\mathrm{dx}^{2}}+3\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}=\mathrm{x}\ell \mathrm{n} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ (iii) $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{x} \ell\mathrm{n}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$ (iv) $\left(\frac{d^{4} y}{d x^{4}}\right)^{3}+4\left(\frac{d y}{d x}\right)^{7}+6 y=5 \cos 3 x$ Answer: 1. Equation (i) : Order = 2, Degree = 1 2. Equation (ii) : Order = 2, Degree = 1 3. Equation (iii) : Order = 2, Degree = 1 4. Equation (iv) : Order = 4, Degree = 3

Step-by-step solution

Equation (i)

The original equation is $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}=\left\\{1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{4}\right\\}^{\frac{5}{3}}.$ The highest order derivative in the equation is the second derivative, so the order is 2. Now, the power of the second derivative is just 1. Thus, the degree in this case is 1 for the highest order derivative.

Equation (ii)

The original equation is $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+3\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}=\mathrm{x} \ell \mathrm{n} \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$. The highest order derivative is still the second derivative. Therefore, the order is 2. The power of the second derivative on both sides of the equation is 1 (Note: we ignore the ln function). So, the degree is 1.

Equation (iii)

The original equation is $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{x} \ell \mathrm{n}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$. The order is again determined by the second derivative, making the order 2. The power of the second derivative is 1. Therefore, the degree is 1.

Equation (iv)

The original equation is $\left(\frac{d^{4} y}{d x^{4}}\right)^{3}+4\left(\frac{d y}{d x}\right)^{7}+6 y=5 \cos 3 x$. In this case, the highest order derivative is the fourth derivative. This makes the order 4. The power of the fourth derivative is 3, so we have a degree of 3 for the highest order derivative. To summarize, the orders and degrees are as follows: 1. Equation (i) : Order = 2, Degree = 1 2. Equation (ii) : Order = 2, Degree = 1 3. Equation (iii) : Order = 2, Degree = 1 4. Equation (iv) : Order = 4, Degree = 3