Which of the following differential equations | vedclass

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(C) To find the order and degree of a differential equation,we first express it as a polynomial in terms of its derivatives.For option $(a)$: $\frac{d

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$\left[1 + \left(\frac{dy}{dx}\right)^3\right]^{2/3} = 4\frac{d^3y}{dx^3}$

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(C) To find the order and degree of a differential equation,we first express it as a polynomial in terms of its derivatives.For option $(a)$: $\frac{d^4y}{dx^4} + 8\left(\frac{dy}{dx}\right)^6 + 5y = e^x$. The highest order derivative is $\frac{d^4y}{dx^4}$,so order $= 4$. The power of the highest order derivative is $1$,so degree $= 1$.For option $(b)$: $5\left(\frac{d^3y}{dx^3}\right)^4 + 8\left(1 + \frac{dy}{dx}\right)^2 + 5y = x^8$. The highest order derivative is $\frac{d^3y}{dx^3}$,so order $= 3$. The power of the highest order derivative is $4$,so degree $= 4$.For option $(c)$: $\left[1 + \left(\frac{dy}{dx}\right)^3\right]^{2/3} = 4\frac{d^3y}{dx^3}$. Cubing both sides,we get $\left[1 + \left(\frac{dy}{dx}\right)^3\right]^2 = 64\left(\frac{d^3y}{dx^3}\right)^3$. The highest order derivative is $\frac{d^3y}{dx^3}$,so order $= 3$. The power of the highest order derivative is $3$,so degree $= 3$. Here,order $=$ degree.For option $(d)$: $y - x^2\frac{dy}{dx} = \sqrt{1 + \left(\frac{dy}{dx}\right)^2}$. Squaring both sides,we get $(y - x^2\frac{dy}{dx})^2 = 1 + \left(\frac{dy}{dx}\right)^2$. The highest order derivative is $\frac{dy}{dx}$,so order $= 1$. The power of the highest order derivative is $2$,so degree $= 2$.

Which of the following differential equations has the same order and degree?

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