The order and degree of the differential equation \((\frac{d^{2}y}{dx^{2}})^{2}+(\frac{dy}{dx})^{2}=x\sin(\frac{dy}{dx})\) are:
A. order 2, degree 2
B. order 2, degree 1
C. order 2, degree not defined
D. order 1, degree not defined
Official Solution
Correct Answer: order 2, degree not defined
Explanation:
The order of a differential equation is the order of the highest derivative present in the equation.The degree of a differential equation is the power of the highest order derivative, provided the differential equation is a polynomial equation in its derivatives.
For the degree to be defined, the equation must be free from terms like $\sin(\frac{dy}{dx})$, $e^{\frac{dy}{dx}}$, $\log(\frac{d^2y}{dx^2})$, etc. (i.e., the derivatives must not appear inside transcendental functions).In the given equation, the term $x\sin\left(\frac{dy}{dx}\right)$ involves the first derivative $\frac{dy}{dx}$ inside the transcendental function $\sin()$.Since the equation is not a polynomial in its derivatives, the Degree is not defined
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